Since "can't wait to" is a common English phrase for "really want to", it would be better (and sound less awkward) to translate "really want to". really: 정말 or you can also use 너무 want to: 싶어요. go back: 돌아가다 means more of turning around from where you are and going back. What you mean (I'm assuming) is to go back and visit again, in which 다시 (again) 가보다 would sound better. when you add 보다 to a verb stem, it means to try that verb, so 가보다 is like try going, or visiting. Also, in Korean subjects are often omitted (unless you are introducing a new subject) Putting all of this together, it would be 정말 다시 가보고 싶어요. If you want to kind of emphasize that you want to go back, you can add 한번 (once) and say 정말 다시 한번 가보고 싶어요. Hope this helps!

"It is still winter up here" will be a complete sentence, where as "still winter up here" isn't a complete sentence. However, considering you are writing this on your insta page (for personal things I'm assuming), there should be no problem. It's only a matter of how formal you want to sound. "It is still winter up here", because it is a complete sentence, sounds more formal than "still winter up here". If you want to make it informal, then "still winter up here" would be better.

I'm going to assume you read this somewhere on the internet or during a chat. 담에 is what people use that is short for 다음에 in chat or online language. 다음 means next and 에 is time/location particle.얘 which is more colloquially short for 이 애, which is short for 이 아이. 이 is like a prefix (prefix probably isn't the correct term) for this like 이것 means "this thing". 아이 means child, but is also colloquially used as person when referring to someone in similar hierarchy like friend. 같이 means together. 안 when attached in front of verbs mean not or don't. For 갈 거임, this 임 is like a speech ending, also more commonly used in chatting. 갈 거 is short for 갈거예요 or 갈것이다 which are future tense for 가다 go. To put all of this together, it means, "Next time, I'm not going with this person" Hope this helps!