Generally, can illiterate characters easily find NPCs who will help them out with small tasks -- writing and reading letters or notices, mostly? Assuming the character doesn't want something very time-consuming, is it all right to just say he or she found someone to help him or her out, or is it preferable to ask a DM every time?

The scribe would be out of business if this were the case.

Literate NPCs would likely be few and far between, and those that are literate would probably have better things to do (like say, running the City) than write letters for escaped slaves and peasants.

See the scribe.

Rather than asking a DM in order to get an NPC to help you, I suggest having your illiterate PC track down a literate PC in-game and pay/ persuade/ intimidate/ blackmail that literate PC into reading notices to your illiterate PC and writing notices on his behalf.