The Modifier space is where you put the Ability Modifer - so with a STR of 14, you would add +2 (page 36 in both "Heroes of.." books.) This box is helpful for powers which have a secondary effect. For example - one attack might also give allies X Temporary Hit points, where X is equal to a certain Ability Modifier.
As mentioned in the other answers, the "Check" space is where you put your TOTAL. This is the value you would use for an Ability Check (using raw STR to open a door), or a Skill Check (Using STR-based Athletics to scale a wall).
The Misc. space is where I usually put special modifiers. Usually this is +5 for a trained skill, +2 for a Racial or Feat Bonus, or a -1/-2 for an Armor Penalty. I don't usually write down the half-your-level math since that is always a part of both checks.
For the Weapon/Powers, if you don't have a DDI and aren't using the online Character Builder - you'll probably want to use those for your most common attacks. Generally, I tend to fall back to writing the actual calculations on the back of my sheet and just keep the front of the sheet clean and readable so it's easy to refer to in game.
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.
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.