We know they need to breathe, since you can hear it when you stay still and you can see them breathing. How long they can hold their breath is unknown, but being terrestrial based creatures it would decrease as they tried to move or fight underwater.
Their lifespan is unknown, though I have the personal belief that if they live long enough they will grow from stage one to stage three without evolving. For fighting underwater, most abilities we've seen would be inefficient.
Rock Throw would be impaired due to the resistance of water.
Charge and Leap Smash wouldn't work will lessened gravity and more resistance.
Fire Breath is useless.
Lightning Strike, Aftershock, and possibly Vortex would quickly dissipate in water.
The invisibility of Decoy would be nullified by water displacement.
Rock Wall is useless when everything can move freely in 3 dimensions rather then 2, as would Fissure.
Lava Bomb would cool quickly, making it essentially a primitive organic cannon ball. Less useful, but still OK.
Acid Spit and Web Snare would dissipate in water, though the webs may last a bit longer if they stick.
Spider Trap would either drown or be useless as everything swam above it.
Mimic would only work if Gorgon could hold its breath as it used it.
Banshee Mines, Abduction, Supernova, Warp Blast, and Tongue Grab would be the most useful if the monster could survive underwater.
And since we know the Destroyer of the Ajax and The Host are out there, there could easily be an aquatic monster. personally, I think Wraith would be the best adapted to fight in water. Its typical floating movement and aerial capabilities with Warp show it's used to fighting in 3 dimensions, and 3 of its abilities still work with minimal interference in water. Also it would look really cool to see one slithering through the water like some sea serpent.
Monsters can survive all kinds of stuff, they probably are already capable of surviving in extreme temperatures or could adapt to it.
And no, no there isn't.
There were artic and lava maps early in development. The artic ones were dropped because it felt weird for hunters to wear the same clothing in blizzards as in deserts, some they replaced them with less extreme ice maps. The lava maps were removed because they couldn't find a way to put lava in and not have it work as a one hit KO, so they replaced it with acid.