"Let's go to the hospital immediately."
へ means "toward" or "in the direction of" and に means "to" you could use へ but it lacks the urgency that すぐ implies
No that's just "let's go to the hospital" the すぐに indicates that you're saying let's immediately go
It's not a separate word, just that they tend to separate fragments regardless whether it is a word or not.
It isn't. It's just separated as part of the question challenge to make sure you're actually reading what you're selecting.