Escalating danger (fun) over time

Some labors may or may not fall into this category, depending on what actual tasks you tend to run in your workshops, and your personal play style.

EDIT: separated "only affects speed" labors from "affects quality but you probably won't care" labors.

Labors that only affect speed

• All hauling tasks. This is what peasants excel at. Nothing beats designating 'dump' on that 30x30 strip mine and watching your peons clear it in a matter of moments. Or furnishing those twenty rooms before the migrants can even get from the edge of the map to the fortress doors.
• Wood cutting can be useful, but you may want to have some spare axes so your peasants don't steal them all and not leave any for your axedwarves. (As a bonus, your armed peasants will stand a slightly better chance if they do end up in a fight.)
• Stone detailing, if you're willing to micromanage a little. Skill is unimportant for smoothing, so you can designate large areas to smooth and send your peasants to it. If you designate engraving, however, you'll want to disable this labor again on your peasants so you don't get ugly carvings everywhere.
• Feed patients/prisoners and recovering wounded
• Wood burning, furnace operation and other "tier 2" ingredient production tasks. Many of these production tasks have set formulae (e.g. 1 bituminous coal + 1 fuel = 3 fuel). Skill has no effect on production except for the time it takes. If you hit a couple good coal veins, it could be very useful to have a bunch of smelters set to make coal into fuel, or ores into bars, set to repeat.
• Plant gathering, if you feel the need, although farming is generally preferable.
• Farming (Planting). Usually one full-sized farm plot and a couple skilled planters are enough to keep a fortress going, especially if you buy any food or drink from caravans. However, it might work well to have lots of peasants with planting enabled, rather than a couple dedicated farmers, but I haven't tried it.
• Cleaning. Very useful when you wipe out a dozen ambushers at the gate and everyone is tracking blood all over the place.
• Pump Operating, although dwarf-powered pumps are generally of limited use.

Labors that affect quality (but you probably won't care)

• Architecture. Skill does not affect building speed. It does have an effect on the quality of constructed buildings, but this is generally very minimal (as outlined in the linked article).
• Mechanics, in some cases. The quality of mechanisms, traps, etc. is not terribly important, unless you really want Urist McLookyLoo to be ecstatic about that Fine Stonefall Trap he saw last week. EDIT: Mechanism quality does affect trap quality, as pointed out by Raven in comments.

Straight forward answer: Yes, they can walk over traps without triggering them.

More detail:

Traps that can be triggered themselves (stone-fall, weapon and cage) will not be activated by your own dwarfs, except apparently when they are unconscious (see DF Wiki).

Spike traps have to be triggered by an external mechanism anyway - pressure plate, lever, etc - and so they could be a threat to your own dwarfs, depending on how you configure them, but won't be by default (unless somehow your dwarfs drop on to them).

All other types of "trap" involve the clever deployment of various mechanism by you (to create flooding rooms, pit falls, etc) and so whether they present a danger to your dwarfs or not is entirely down to how well you design and build them.