Sure. If you expect to have 120V everywhere, that is. If you're into boon docking, remote campgrounds etc. you might want to look for a TV that can run straight off 12V DC, inverters are not very efficient.
yes you can get a cheap TV. I have seen 24" units at Walmart for like $79... unless you are close to a city the antenna on your trailer may be the weak link to picking up stations if you are remote to a big city. Of course if you are only looking at doing a DVD while it rains outside... then it is easy.
We have several 'regular' tvs in our rig. They hold up just fine.
Scott and Liz - Southern NM
2012 Wildcat Sterling 32RL - w/level up (best option ever)
2007 Chevy 2500HD 4dr short bed Duramax w/allison
Reese Fifth Airborne air ride king pin coupler with Sidewinder