Saltwater can ceranly be stored but it is recommended to always have an operating waterflow pump or airstone going to keep the water moving..you don't want it to stagnate. As for a cover, a friend of mine left hers open (she used a 45 gallon trashcan for her RO
saltwater) but she also had it in a closet so no dust or anything could get in. If it is open air in the house, you are chancing things like dust or chemicals, like from dusting spray or air fresheners, contaminating the water. Room temp is fine and if it is only a degree or two lower than your tank temp, ther s no reason to heat. HTe mixing into the "normal" temp will bring it up enough that it won't stress things.
If you don't have standard water containers, I used to use a Rubbermaid tub. I used a low level powerhead and snaked the cord out of the tub over the handle. I could still lock the lid down.