Nothing is water proof, only water resistant at best.

A water "proof" watch for example, might be rated to 30m but diving into the water or diving down some depth and furiously waving your arms around can let water in... it's about the pressure.
Same as getting in the shower with a water proof watch is a bad idea, it may be protected against the water, but not the steam, which can still get in.

So, it can only be water resistant and 99% of the time, is something that needs attention and/or maintenance.

Brushed motors can and will run under water, some people even wear the brushes/comm in this way.
Just make sure you re-lube the bearings/bushings after.

Brushless motors is a different topic, some people do, some people don't.
The newer Velineon motors are apparently sealed so these can be ran in water but again, only water resistant... nothing is proven against water.
Most of the thing's I read suggests that the main reason why brushless motors are not ran in/under water is because it may prove difficult to get that water back out, especially in in-runners.
Out-runners may be a different story, I'm not sure to be honest.

As for electrics, you can use plastidip (I think that is the right name) which you can either paint or dip your electrics and puts a film over it all giving it another layer of protection against water.

The biggest no-no that I know of with electrics and water is salt water.
Salt water has a higher conductivity and would make things short out much much easier.

Clean fresh water should be fine for just about everything, dirty water and mud is a bit bigger risk but salt water is the biggest killer and makes metal rust a whole lot quicker.

Quite a few people run all sorts of different batteries under water, so I am not sure about that one.

If I am wrong about anything, please correct me, I would hate to be giving out bad advice.