Listen to what Ka-pi said and get bank. It's $5 USD and lasts for a year, and even if you let your sub run out, it keeps your stored Pokemon in the meantime. Anything you want to transfer over to your new game, you can do so at your leisure.
As far as how to tell if a mon is hacked, there are definitely ways you can tell. The online checker looks for these things: illegale stats, illegal moves, and symbol representing what game (aka region) the Pokemon was born/generated. So long as you don't have a Pokemon with stats that can go past the max at its current level, a move it can't possibly ever learn (event Pokemon complicate this a little), and doesn't come from a region that it couldn't possibly have been available, you should be fine. You can go to Bulbapedia and look up all this info for any mon, and I'm sure there are online sites that let you plug in your mon's info and have it tell you if it's legit.
Otherwise, the online checker is notoriously easy to beat. Word of advice: the people wonder trading you shiny legendaries and battling with a team full of six shinies with maxed out stats and egg moves - they're hacked. Most people are either using hacked mon, or using hacked mon to breed legit ones (that's what I do, 6IV Ditto here). To get caught for hacking and banned is equivalent to tapping a police officer on the shoulder, knocking coffee out of his or her hand, then punching a woman in the face. You have to really be trying.`So in all likely hood, those mon you inherited came from hacked means in some way, but I bet they are fine. If you see a Charizard with Hydro Pump or something, then that should be setting off red flags in your head, lol.
Also, the thing you REALLY want to be careful about is not losing rare items. PokeBank does not allow you to transfer items. There was an even early in Y that gave people access to Blazikenite (Blaziken's mega stone) that was not originally in the game, and there is no other way to get it if you lose it, other than trading with someone else. You don't want to lose that, so if you do restart your game and you have it now, make sure to ask someone to temporarily hold it for you. Same thing with either Pinsirite (Y comes with Heracrossite, X comes with Pinsirite).
And certain berries could only be self-obtained by playing certain mini-games online that are no longer available, like the Custap Berry. Definitely have someone hold onto that for you if you have it now. I keep trying to end this post, but I keep thinking of more things, lol.
