It sounds like you're asking which rolls gain the ubiquitous +1/2 level bonus. The easy answer is:
Most every d20 roll that gains an ability stat modifier as a bonus also gains 1/2 your level as a bonus to the roll.
- Attack rolls, initiative, skill checks, ability checks
Rolls that don't use a d20 don't get the level bonus.
d20 rolls that don't get an ability modifier as a bonus don't get the level bonus either.
All defenses also gain +1/2 level, so as to scale with attacks.
There are, of course, always edge cases, but the above is a good rule of thumb for 1/2 level bonuses.
Now, your other questions...
The "Check" box is for ability modifier + 1/2 level.
I personally find this produces more math than it's worth because then you have to subtract the level bonus when recalculating damage and the like, so I put just the ability mod in it.
"Misc." is for "Miscellaneous"
Extra bonus from feats, items, and so forth, go here.
"Add +1 to the modifier" means increase a +2 to a +3, and your attack increases by +1.
The +1 in their instruction is the +1/2 level bonus at level 2. If that counts as recalculation then yes, recalculate your attack.
Storm Hammer example
Attack: Wisdom vs. Fortitude
This means that you roll 1d20 + your Wisdom modifier + 1/2 your level + weapon proficiency (because Storm Hammer has the 'weapon' keyword) + any additional bonuses from weapon enhancement or feats or the like.
(When Wizards mentions an ability as part of a power or feature, like "Wisdom" in the attack line here, they mean the ability modifier. If they meant the whole stat, they'd have said "Wisdom score.")
If the result you get from the attack roll meets or breaks the target's Fortitude defense, you hit! So move to the hit line.
Hit: 1[W] + Wisdom modifier lightning and thunder damage.
Level 21: 2[W] + Wisdom modifier lightning and thunder damage.
That's 1[W] (the damage die your weapon deals, in your case 1d8) + your Wisdom modifier + any additional bonuses from weapon enhancement or feats or the like. The total of that roll is the damage you deal to the target of the attack.
The modifier you add to a roll only increases when one of those subcategories increases: If your Wisdom score increases, the modifier might also; or you could take a feat that gives you a bonus, or get a better weapon, and so forth. This means that your attack rolls will increase by at least 1 every other level (the +1/2 level bonus increases) even if you don't do anything else to bump them up. Your damage will probably increase more slowly, from feat and item drops, and as it says on the card at level 21 you get to roll two weapon dice instead of one.
Better Resources
All I can really point to is the Player's Handbook 1 or Heroes of the Fallen Lands. They'll walk you through this process and explain these mechanics much better than I can, in a scope far beyond what a question like this can cover. The Rules Compendium is also invaluable, but doesn't walk you through this process.
A crushing wall trap doesn't require an attack roll (2000 DMG 115) and neither does the compacting room (2003 & 2012 DMG 72)—and there's no provision in either DMG for stapling spikes on the wall to make either traps nastier— and while the DM makes attack rolls for a spiked pit trap (20 ft. deep) (2000 DMG 115) and a spiked pit trap (2003 & 2012 DMG 72), these aren't normal attacks. Further examples are necessary even to have a question because…
DR Only Affects Damage from Weapons and Natural Attacks
A hero can kill a creature he can't damage with his weapon by luring it into a nasty enough trap, and a creature with up to 6 hp and DR 5/magic can be killed by an unlucky fall.
Damage reduction hasn't affected anything but damage from weapons and natural attacks in Dungeons and Dragons, 3rd Edition (2000) and Dungeons and Dragons 3.5 (2003). This means most traps (except those employing weapons, which some do) will automatically overcome a creature's damage reduction.
Therefore unless the trap's an actual weapon or natural attack, DR doesn't apply. Moving walls (unless your poor PC is being beaten by a creature using a wall as an improvised weapon) and falls will bypass DR. The spikes at the pit's bottom are the DM's call (cf. armor spikes/spiked armor), however. (Don't forget to totally swoop on them, Grayhawk style, if the DM is putting +1 spikes at the bottom of pit traps just to overcome a PC's damage reduction; you earned those!)
"Why Would Damage Reduction Work This Way?"
I don't know. Seriously. I'll happily speculate, though. Although I'm unfamiliar with earlier editions of Dungeons and Dragons, in at least Advanced Dungeons and Dragons and Advanced Dungeons and Dragons, Second Edition some creatures were flat-out, straight-up immune to weapons if the weapon didn't have a sufficient magical plus (e.g. Advanced Dungeons and Dragons has in its Monster Manual the vampire, which has the entry Special Defenses: +1 or better weapons to hit (99); Advanced Dungeons and Dragons, 2nd Edition has in its Monstrous Manual the pit fiend, which has the entry Special Defenses: +3 or better weapons to hit (11)).
This led to the phrase You must be this tall to play; that is, a party must possess sufficient magical equipment (which was much rarer in earlier editions) to combat a particular monster else the party, upon encountering the monster, fought it with magic spells (which were much more constrained in earlier editions), parleyed (not usually to the party's benefit), fled, used their combined ingenuity, were granted an out by the DM, or just died. Since all but the first were… distasteful, and the first unavailable to many characters, the concept of +X or better weapons to hit was changed to damage reduction in Dungeons and Dragons, 3rd Edition, which let combat ensue even against monsters who would've formerly been invulnerable to the party's warriors' attacks, albeit at a disadvantage if the party's warriors lacked the proper gear.
With that in mind, damage reduction wasn't ever intended to be a panacea against damage in general and was instead developed as a legacy to evoke older editions which had monsters that could only be hit by +X or better weapons, and weapons only and specifically. So, yeah, my guess is that falls and traps kill monsters with damage reduction in Dungeons and Dragons, 3rd Edition and its update because those killed monsters in older editions, too.
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A Comparison of DR in Dungeons and Dragons, 3rd Edition (2000) and Dungeons and Dragons 3.5 (2003, 2012)
Although tagged as a Dungeons and Dragons, 3rd Edition question, it's useful to know these rules aren't edition-specific.
The Player's Handbook
The two big deals for damage reduction in both editions of the Player's Handbook are the barbarian's damage reduction and the damage reduction granted by the spell stoneskin.
The Player's Handbook (2000) describes the barbarian's damage reduction as
the extraordinary ability to shrug off some amount of injury from each blow or attack. Subtract 1 from the damage the barbarian takes each time he is dealt damage. (25)
The Player's Handbook (2003, 2012) describes the barbarian's damage reduction almost identically, except it clarifies further: The barbarian
gains the ability to shrug off some amount of injury from each blow or attack. Subtract 1 from the damage the barbarian takes each time he is dealt damage from a weapon or natural attack. (26)
Further, the Player's Handbook (2000) contains the 4th-level Sor/Wiz spell stoneskin [abjur] (257), which reads
The warded creature gains resistance to blows, cuts, and slashes. The subject gains damage reduction 10/+5. (It ignores the the first 10 points of damage each time it takes damage, though a weapon with a +5 bonus or any magical attack bypasses the reduction.) (257)
And the Player's Handbook (2003, 2012) describes the spell stoneskin almost identically, except it clarifies further:
The warded creature gains resistance to blows, cuts, stabs, and slashes. The subject
gains damage reduction 10/adamantine. (It ignores the first 10 points of damage each
time it takes damage from a weapon, though an adamantine weapon bypasses the reduction.) (285)
Emphasis in quotations mine.
The difference here, then, between damage reduction in Player's Handbooks for Dungeons and Dragons, 3rd Edition and Dungeons and Dragons, 3.5 is that—y'know, obviously, apart from how damage reduction works—is that when using the Player's Handbook (2003, 2012) one no longer needs to look for confirmation in the Dungeon Master's Guide or Monster Manual to determine absolutely that damage reduction only works against weapons, even though that was very strongly implied by the Player's Handbook (2000), as shown by the emphasized text.
The Dungeon Master's Guide
Although it received the most serious overhaul during the Dungeons and Dragons, 3rd Edition transition to Dungeons and Dragons 3.5, the description of damage reduction in the Special Abilities section of the Dungeon Master's Guide (2000) reads
Some magic creature have the supernatural ability to instantly heal damage from weapons or to ignore blows altogether as though they were invulnerable. The number in a creature's damage reduction is the amount of hit points the creature ignores from normal attacks. (73)
and the Dungeon Master's Guide (2003, 2012) reads
Some magic creature have the supernatural ability to instantly heal damage from weapons or to ignore blows altogether as though they were invulnerable. The numerical part of a creature's damage reduction (DR) is the amount of hit points the creature ignores from normal attacks. (73)
Emphasis mine. The remainder of descriptions of damage reduction in the Dungeon Master's Guides is how damage reduction's overcome by weapons (or "normal attacks") and natural attacks.
The Monster Manual
The Monster Manual doesn't say damage reduction applies to anything but weapons and natural attack either.
According to the Monster Manual (2000) damage reduction means
The creature ignores damage from most weapons and natural attacks. Wounds heal immediately, or the weapon bounces off harmlessly (in either case, the opponent (in either case, the opponent knows the attack was ineffective). The creature takes normal damage from energy attacks (even nonmagical ones), spells, spell-like abilities, and supernatural abilities. A magic weapon or a creature with its own damage reduction can sometimes damage the creature normally, as noted below. (9)
The Monster Manual (2003, 2012) is almost identical, saying that damage reduction means
The creature ignores damage from most weapons and natural attacks. Wounds heal immediately, or the weapon bounces off harmlessly (in either case, the opponent (in either case, the opponent knows the attack was ineffective). The creature takes normal damage from energy attacks (even nonmagical ones), spells, spell-like abilities, and supernatural abilities. A certain kind of weapon can sometimes damage the creature normally, as noted below. (307).
Emphasis mine.
And then the Monster Manual (2000) describe how damage reduction is overcome by magic weapons with different bonuses, weapons composed of different materials, and certain types of creatures, while the Monster Manual (2003, 2012) describes how damage reduction is overcome by weapons composed of different materials and certain types of creatures. Because while what overcomes damage reduction changed between Dungeons and Dragons, 3rd Edition and Dungeons and Dragons 3.5, nothing changed about damage reduction itself.
Possible Contentions & Other Sources
A bone can be picked with the phrase normal attack, which is used instead of weapon or natural attack in some discussions of damage reduction, and that is somewhat problematic (e.g. in the Dungeon Master's Guide (2003, 2012), the CR 2 trap spiked pit trap (71) has spikes that get an attack roll, the CR 4 trap collapsing column (DMG 72) gets an attack roll), but I think that it's possible to agree that falling into a spiked pit or being crushed by collapsing column, for example, isn't a normal attack.
Another can be picked—for several reasons—with the frequent, ill-advised, and formally undefined use of the word blow. Dungeons and Dragons, 3rd Edition and Dungeons and Dragons 3.5 commonly and most frequently uses blow to mean an attack from a weapon or natural attack. Restrain your inner 12-year-old while I list feats like Awesome Blow (2003, 2012 MM 303), Death Blow (SF 6), Low Blow (Rac 166), Telling Blow (PH2 83), and—undoubtedly present due to lax editorial oversight and renamed by any player whose character took it—Toothed Blow (Sto 94); the 9th-level Tiger Claw martial maneuver feral death blow [strike] (ToB 87) and the 5th-level Diamond Mind martial maneuver disrupting blow [strike] (ToB 63); and even the 3rd-level Sor/Wiz spell dolorous blow [trans] (SpC 70). All of these involve weapon attacks or attacks with natural weapons.
Saying the SRD is incomplete and lacks context about this issue is a little unfair. The SRD omits only the examples from the Monster Manual (2003, 2012) for damage reduction, and all of those Monster Manual examples mention weapons or natural attacks, and, in fact, seem to be very careful to do so.
Finally, damage reduction is far more vague in Pathfinder, with its change to damage reduction's language, but that's beyond this question's scope.
Best Answer
Spells per day
Spells per day is based on your character class, character level and primary stat. Let us take a wizard for example (primary stat is intelligence), at level 1 he has 3 level 0 spells and 1 level 1 spell, our wizard has an intelligence of 16 which would give him a bonus level 1, level 2 and level 3 spell. He doesn't get the last 2 yet since he does not have access to those yet.
Concludingly our wizard has 3 level 0 and 2 level 1 spell (1 from wizard + 1 from intelligence).
Spell save
GMJoe allready pointed this out in the comments but the DC for a spell is simply a 10 + the spell's level + your primary ability modifier (intelligence for our wizard), resulting in a DC 14 for a level 1 spell (10 + 1 + 3), this would mean filling in a 14 in the spell save DC column (left of 1), 1 for spells per day (level 1 ) and 1 bonus spell. I would put a 3 in the DC mod as that is your intelligence modifier (altho I have never even looked in that box myself).
I assume that the level 0 values are easy to solve yourself now?
ADDENDUM
Special cases
KRyan's comment brings us to to field of non core classes. I used a wizard as example in my answer but some classes use different stats than one might expect. The favored soul for example is a cleric like caster class that uses Charisma instead of Wisdom for bonus spells but still uses Wisdom to determine spell save, in effect, she has two primary stats being Charisma and Wisdom.
Summary
The key point is to check what the primary stat or stats is/are for the class of your choice.
The majority of the classes use the same stat for spells per day and spell save (i.e. Wizard, Cleric and Druid) while some classes use a different stat for spells per day and spell save (i.e. Favored Soul)