In Time Warner Cable’s FAQ online they have:
Question:
Does Time Warner Cable have seasonal/vacation hold?
Answer:
Yes, in some locations. Please contact us to see if your area qualifies.
Maybe your cable company has the same.

Regarding other utilities, I guess it depends how long all these repairs will take. It sounds like major damage, so if you are going to be gone for more than a week or so, you could probably make the case that you wouldn’t be using your normal amount of electricity and they should be covering more. The other utilities (water and phone) shouldn’t be used much without you there.

I don’t know anything about asbestos, so I’ll others respond to that aspect. You might want to ask the HOA if there is anything they can do to make sure things are done properly.