"How stable" is pretty much a subjective judgment. Suggest you investigate screwing/bolting the TV to the stand. My LCD had a hole in the base for that purpose. Another alternative a strap from the back of the TV to the base, and/or to the wall. My LCD also had a provision for this and included...
It works great for my wife. When we went HD-HT she went bananas. The idea that she could punch a button on the remote that pulled up a function (Watch TV) or (Watch TV HT) and everything falls into place made a true believer of her - and a great supporter of HD
You could try this: There is an excellent do it yourself antenna on youtube. Unfortunately the powers-that-be won't let me post it yet maybe look at you tube then slashwatch?vequalsignEWQhlmJTMzw
Now that you have stopped assuming and checked it, what did you find. What trim? I already suggested running it under the molding (trim?) at the bottom of the wall. BTW, my carpet is not glued to my slab either
What is on the floor for a covering? You can lay the wires under it, adjacent to the tack strips. Is their molding along the wall/floor junction? You can raise it slightly, or hollow out behind it. Worked for me. YMMV
The powers that be won't let me me help you directly. Google something like: TV screen size equivalent and see what you get. You would like a pge like that which josefDOTorgSLASHtvcalcDOThtml contains