As far as I know storing out of state firearms in RI is not illegal, but I'm not positive. I do not know of a place that you could store them at other possibly at a gun shop that you trust.
There are good parts of Providence, but they are definitely higher up on the price range. The houses you have looked at online that are less expensive and "really bad" as you put it are cheap for a reason. Bad neighborhoods are abundant in Providence. If you do want to buy a multifamily and rent out half of it, search the surrounding towns and cities. You may be paying a little more, but you will be a lot better off in the long run if you stay out of Providence.
You will only find central air conditioning on new houses. The vast majority of older homes have to rely on window units. Some homes are probably retrofitted with split systems, but those will definitely be on th upper price range and scarce.
Did you look for multi-family homes in Cranston, Warwick, West Warwick, North Providence, East Providence and other centrally located communities? There might even be some in East Greenwich. Providence definitely has the most, but I would never live in Providence even if my home were bought and paid for.