I get cold water out of the tub. Yes my home is on a slab. Seems like the cold water valve for the sink is defiantly acting odd. No turn know was on it and due to the corrosion I can't tell if it's on or off.
So more than one problem. Sure, you will want new stop valves, and when you post the photo of the current stop valves, you will get some info.
Correct me if I am wrong, I think you are saying that you get no water fill the toilet unless you turn on the faucet at the lavatory to warm. If that is the case, the path for the cold water is blocked, and I think you will be filling the toilet with hot water that may have cooled down some.
Search for a plumbing access panel. With copper in slab, your connections are out of the slab like this (this photo is not related to CrimIsGrim and is only for illustration):
You can get plenty of cold to the tub. Because you ran the cold for several minutes, you know it is not just cooled-down hot water, right? Is there maybe access behind the tub faucet, maybe in a linen closet? Often the same 3/4 inch pipe will supply all of the cold for the bathroom. If that is your case, somehow the cold water between the tub and (lavatory+toilet) is blocked.
How old is this home
? Did you have a home inspector? Can you ask the former owner for info?