Yes, you can omit "any" in all those examples. Side note: If you are having dinner and there is a basket of bread, the normal think to ask would be, "Would you like some bread?" the formal way to ask is: "Would you care for some bread?"

In the USA, you would never hear a native speaker say "since two months ago." One would say, "I've lived here since April" or "I've lived here for two months."

It is just fine :)

"One who can better help you" would be better. :)

This isn't something you would normally say in regular conversation, although one could. It means, life is dull (boring, unexciting, gray, bland) when your miserable. It implies that it is hard to feel anything, enjoy anything, see beauty, etc. when you are miserable.

Generally speaking, it means "for the sake of it", "for fun". You wouldn't want to answer a serious question with it. For example, Boss: "Why did you leave early yesterday." You: "For the hell of it". <<< Do NOT say that!! I most often hear it when someone makes a suggestion. For example, "Just for the hell of it, let's try (insert potential solution to an engineering problem) and see if it works".

Interesting discussion.. you could get away with either in normal conversation and writing, but "care" is the most correct answer, as noted below. Now, you could change it to, "We take every precaution..."

If something is a "sure thing" then it is definite, it can be trusted, it is certain. If you ask someone to do something, and they say "sure thing", they are enthusiastically saying 'yes, I will'.