Do the items offered from a store depend on the character in any way (e.g. on the character's level), or only on the merchant? Also, do the act and the difficulty level play a role?
Act and Difficulty seem to be the biggest modifiers for item quality in Diablo III - Act IV Normal merchants will give you Normal-grade items around req 21-25 even if you're level 60, while Act I Inferno will give you high Hell or low Inferno grade items with requirements in the late 50s. The level of the character starting the game does appear to influence it slightly to about a 5-10 level degree of potential variation - join up with a higher level character on the same difficulty as you and check out their Act I merchants, and you're likely to see higher leveled items - but in general you're going to see better vendor items in later Acts and on higher difficulties.
As for what types of items they stock, that depends on the merchant. The fence in the back of New Tristram will always have a couple rings and amulets for sale, no matter when you visit him.