Will items in your home despawn?

No, they will not. The in-game help guide even mentions this:

After purchasing a house, you can safely leave your items inside — They will always be there when you return.

I've tried this with a lute and a bee in a jar and so far so good :)

I also dropped some stuff at a shop and it is still there. I also dropped a few books and a dagger on the street and I guess the people of whiterun don't like books and a weapon cause it is still there to this day.

I find that whenever I drop something in a store, it is given to me by the store owner. They say something like "Hey, you dropped this. I figured you wanted it back," and then it says item(s) added and it's in my inventory.

I don't think they will disappear, but when I place my items in my home and return, I find them everywhere but where I put them.