the 11.1v will push the gun faster, and the extra wear can cause issues like pme etc that will kill the box.
however if you dont want a rate of fire increase then just get some larger 7.4's, you want a larger mah rating (milli amp-hours) for longer run time, generally i find over ~1100 is generally sufficient for a days shooting if you're not so trigger heavy but as long as the battery fits in the gun there's no harm in having even more capacity.
when it comes to barrel/hop really you're looking at 2 areas.
a good hop will be able to spin heavier ammo, and be a bit more consistent, heavier ammo of course will provide additional range (but a hop upgrade on it's own won't) if it's set up right it'll reduce curve deviation to the left/right (consistent left/right deviation means the hop isn't square in the barrel or the gun's being held at an angle). there's different types available, flat hops can be cheap and effective, although can have a very sensitive "sweet spot", PDI's W hop i quite like as a simple drop-in although it won't handle really heavy ammo, there's the likes of maple leaf's macaron, which are good although the feed lips can cause jamming and need some sanding down. the king is a well set up Rhop, but they're a pain to get right.
a barrel will decrease linear deviation, there's arguments about tight versus wide bore etc but imo this is secondary to quality, although tightbores will need to be cleaned more regularly. personally i get good results from PDI, although i've heard good stuff about EDGI/Orga barrels too.
