Simply put, it means: Isn't this delicious?

For the first sentence, normally people say "shake off this cold". I've never heard anyone say "I can't seem to shake this cold." For the second sentence, I would also lean to "shake off". But in this case, "shake" is acceptable as well.

Not withstanding xarmanla's comment that technically it's the engine that consumes the fuel [well, actually, the ignition of the fuel within the combustion chamber ;)], if you are asking whether the sentence sounds natural, I would say: Cars need/use gasoline in order propel forward.

Both are OK, depending on what you are trying to say or emphasize. For the first sentence, unless you want to emphasize that you are in the process of coming to a certain place, the below would be a bit more natural: You're not going to believe where I just came from.

I would suggest: That must have been fun. If I had gone, I would have been able to see it.

You probably should say: 이태리어 공부하는 한국분을 만나면 좋겠어요. Your original statement seems to have "이태리어 공부" as the subject: The "Italian Study" would like to meet Korean people.

Either is acceptable. There is no difference in meaning or connotation whether you use or not use "up" in this sentence. Purely stylistic.

This would be more natural: A) Did the doctor say that I have to stay home tomorrow? B) No. But she recommends that you do.