No-no! It is the other way around!
"Я смотрю в телевизор" is "I am looking inside the TV set"! You can use it only when:
1) You are repairing your TV set: you have opened the back cover and you are looking trying to figure out what is wrong inside;
2) You are trying to watch a movie, looking into the TV screen, but do not understand what the movie is about and what is going on on the screen (this usage is common in colloquial speech), but I would not recommend to you using it since there are more nuances. It has some negative connotation BTW.
The only way to say "I am watching TV" in Russian is "Я смотрю телевизор" (with no preposition).
It is correct!
I think you're right. But could you clarify:
How are those sentences different in English? (I am looking at you and I am watching you). I am afraid I do not understand the difference well.