If I use the items sitting on shelves (i.e. grand soul gems & alchemy ingredients) will they respawn?

They are NOT in containers they're just sitting on shelves. It looks like a merchant shop.

It does and it doesn't. I still can't figure out if it is suppose to or not.

I don't know if they replenish themselves but I've had things get tossed all over the room, only to magically move themselves back into place.

I've got Honeyside, and I haven't touched anything yet besides taking a potion or two.

With Proudspire Manor, I took every single alchemy ingredient in every shelf. That was over three months ago. They haven't been replenished yet.

My experience is that they never respawn. Therefore I treat everything, no matter how temporarily useful, as a decoration. Except for the Black Soul Gem in Hjerim. Gotta have those. At any rate, after becoming Archmage I ransacked the AM quarters for everything in it, thinking that at least the plants would respawn. Nope. The place looked like an abandoned warehouse after that. So instead of learning from my experience I ransacked Breezeome of everything worth a septim and the same thing happened.

Once you become Archmage that stuff doesn't respawn, however if you don't mind stealing you can regularly loot his room of stuff if you do not do the questline as it does respawn prior to that.

Player housing, i'm pretty sure nothing respawns (I assume that includes the garden in Riften house), though Hjerim in Windhelm is so buggy it's hard to tell with that house!

Inside the house, nothing will respawn - that's the nature of safe locations. The garden will respawn; it's tied to Riften's respawn, and requires you to avoid the city (including the house) for 30 game days.

Thank you.

Decorations they will be.

The Archmage's Quarters are the same way. The stuff you placed yourself will still be there, but the other stuff will respawn on top of it.

The Black Soul gem in Hjerim is the only item I've seen respawn. I'm on the PS3 and it's respawned at least twice so far. It would be nice if once you were to use up all the items that come with the houses the parasitic house carls would at least go out and restock for us!

I've been the A-M 3 times, the quarters have respawned at least once on each playthrough (twice on my current).
The last time was just a short while ago, I took time off to do the Companions from joining through Purity.
I went back every now and then to store a few things, but stayed away more than usual as I wasn't doing much looting.

When I first become A-M, I always clean out the shelves (except for food), the wardrobes, chest, safe, even the display cases and the bin with the leather strips. Whenever it respawns, all of those items are replaced. However everything I have stored in barrels, sacks, wardrobes, chest, and safe, remains intact. (There's one barrel I don't trust/use.)

I'm sorry I don't know about Riften. I buy all the houses but use the A-M as home base, Whiterun for smithing.

I've had everything respawn in Honeyside numerous times over eight plays...