As for your broken TV, all sentences are correct and are emphasizing different aspects of your problem. Let me give some loose translations and try to illustrate the differences.
My TV is broken, so I can't watch TV.
The progressive tense emphasizes the ongoing state of "being broken". You intend to repair your TV, but in the meantime, you can't watch TV. You can't watch the episode you wanted, but you don't sound too troubled by that.
My TV is broken, so I haven't been able to watch TV [or rather anything].
Again, you intend to repair your TV, but now the progressive tense in the second half of the sentence means that you also feel the length of "not being able to watch". (You can't watch anything while your TV is broken.)
I can't watch TV, because [whenever I try to watch TV] my TV breaks. or
I can't watch TV, because [I think] my TV is going to break.
This and the next are a bit strange, because the plain form of 壊れる means either that "every time you watch TV, your TV breaks" or that you're worried that your TV will break. (Cf. 壊れるから、触らないで! "Don't touch, it's gonna break!")
My TV always breaks, so I haven't been able to watch TV.
Again, your TV either breaks every time you try to watch (strange), or you're worried it does (also a little strange), and you're troubled by the fact that you can't watch anything.
My TV broke, so I can't watch TV.
Your TV broke and it doesn't sound like it's something that you're going to (try to) repair. As a simple consequence, you can't watch TV.
My TV broke, so I haven't been able to watch TV [or watch anything].
This is the most hopeless of them all. Your TV is broken beyond repair, and as a consequence you lost your link to civilization. [Replace this by a more reasonable interpretation, by analogy with the above.]
There are other reasons for choosing one tense over the other. For 壊れる one is more likely to say 壊れている, because 壊れた sounds very careless, in the sense of "broken and nothing I'm going to do about it". 壊れている sounds more like you're trying to get it fixed. (Try it with 車 or トイレ!)
(2) is the only real option. いつも閉まる, as in (1) and (3) would mean "always [opening and] closing" (or "always closes [whenever I look]") which doesn't make any sense. And (4) would be "The window is always closed, so he might travel soon." Why would someone close his windows days or weeks before travelling?