I read somewhere it actually speeds up the leveling progress, because you do more (dmg, gold from potions, etc...) at lower lvls. Cannot confirm this with a source though.
This is true for alchemy and speech, because your gain is based on the item's value. Pickpocket is also based on item value, but the value isn't increased with higher skill.
It probably does for alchemy as expensive potions give more xp and good gear can raise your skill more than 100 points.
Not for smithing as the skill gain is the same for everything.
not sure if damage matter for weapons or it's just number of hits.
Smithing increases at the same rate regardless of the item's value (hence the iron dagger debate). Damage doesn't affect your weapon skill gain, but damage taken does increase your armor skill gain.