so the batteries died before they were put away or have you only just noticed coming out of storage?
basically a lot of airsoft pews have very very basic circuitry in them- there's a switch which is the trigger, a fuse, and the motor.
a mosfet is an electronic switch which, depending on the make and model, will offer a bunch of different features.
in this case, a mosfet with a battery cutoff will make the pew much easier to use (as it'll just cut the power when the battery gets too low so you can't kill the battery by over-discharging), it'll also preserve the trigger contacts making them last much longer.
others may chime in with their own suggestions, but something like this:
https://gatee.eu/products/programmable/nanohard
has a battery cutoff
others may have better/cheaper recommendations from the other brands.
of course installing one isn't a trivial thing, requiring disassembly and some soldering skills, so getting a tech to install it would be the best option- folks on here might be able to suggest somewhere good based on your location.
as i mentioned previously you can get away without this, but i wouldn't suggest it for a kid who's not had much experience with these things.
whilst i've heard of this being done, it's not something i'd wanna try even myself, even if you save it you'll end up with a weakened battery.
yes (with a caveat).
the rule of thumb i generally go by is 1C charge rate, so whatever the mah capacity is in ma (in this case 1.3a), this gives a 1 hour theoretical charge time (bit longer in reality for the balancing stage).
batteries can sometimes take more than this (the rc guys for example tend to want faster charge rates when they're out playing) and the batteries can be rated to take it, although it does wear them out quicker.
for airsoft purposes though charge time can simply be solved by more batteries on the day and being organised enough to remember to keep them charged, so no benefit to pushing them harder.