So just to confirm, all wild ingredients (ones that are not shelved or stored, but are found in the wilds) can respawn, right? If so, does this include Nirnroot? Is it confirmed that it respawns? I've been told that it does not.

Finally, what about non-wild ingredients (found stored in homes, shops, and dungeons)? Do they respawn as well?

I need to know this because as a dedicated alchemist, I'm weary of quests that require me to hand over ingredients, especially the blackbriar quest where she asks you for 20 deathbell, 20 nightshade, and 20 nirnroot. I don't want to give away all those ingredients if they won't ever come back.

Yes all plants outside respawn their ingrediants.

The ones in sacks etc, I am unsure about.

Nirnroot's again I am unsure. Although unless there is another Nirnroot challenge I would expect that they do.

There are some respawning samples of Nirnroot. You don't have to worry about that.