when you stay at an inn and theres a chest will it safely store your items?

probably not...
why store them for one night anyway?

I dont think they are. Not sure if there is a time limit, but they wont stay there long. The only safe ones are the ones inside your property and even then there are rumours of glitches that lose all your items.

I use gold weight,lockpick weight,and am waiting for an arrow weight mods.

I don't use mods, so I don't have your problems, I read somewhere that there's a shack with safe storage, try that

The Alchemists Shack near Ivarstead has safe storage.
I have read that Meeko's Shack also has safe storage.

The room that you get when you join the college of winterhold has a bunch of containers that I have found thus far to be safe.