Backyard Aquaponics
A place to discuss aquaponics

Can you put plants that grow under water into the system to oxygenate water as well as give smaller fish such as mosquito fish an area to live and hatch?

Sure you can, i think the main reason people don't is because they compete with harvestable crops for nutrients, and for most people AP is fruit and veg oriented, and most people don't breed in their systems so no need for hiding places.

Also, some fish like goldfish and tilapia might eat them up pretty quickly.

And i don't know how much oxygen plants will provide, but i imagine it would be pretty minimal compared to airstones or gushing water... i certainly don't think it could replace them as an oxygen provider for the system.

no problems, Danny covered it well above.

Vallisneria in pots with rock/gravel works fine.
If you have an excess of N in your system it will help out.

plants do use oxygen at night but it is not that big a deal unless you are high stocked.
(in which case just run some air or ensure good pump circulation).

If you have silvers in there they will love them