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.

It can, or the cutting part wouldn't work.

The Pathfinder rules are based on a list of things characters can normally do, such as using skills, or attacking. This list is not absolute. A spellcaster can also cast spells, or you can't use diplomacy on someone trying to kill you at the same time. However, any deviation from the general set of possibilities needs to be explicitly mentioned in the rules.

Swallow Whole makes no mention of any sort of restrictions on the character, apart from him being grappled. Of course, being grappled itself imposes restrictions on what you can do, and the grappling rules are clear on what you can do if you are grappled, including

Instead of attempting to break or reverse the grapple, you can take any action that doesn’t require two hands to perform, such as cast a spell or make an attack or full attack with a light or one-handed weapon against any creature within your reach, including the creature that is grappling you.

Also take note about the description of cutting yourself free from Swallow Whole:

A swallowed creature can try to cut its way free with any light slashing or piercing weapon (the amount of cutting damage required to get free is equal to 1/10 the creature’s total hit points), or it can just try to escape the grapple. The Armor Class of the interior of a creature that swallows whole is normally 10 + 1/2 its natural armor bonus, with no modifiers for size or Dexterity.

This is not a special type of action only available to swallowed characters (Note how it doesn't say things like "as a standard action"). It basically just states that instead of just hacking away at the monsters insides, you can use some weapons (i.e. light slashing or piercing ones) to cut more precisely, in order to cut yourself free.