1. No, it can't be replaced. I'd understand what you wanted to say, but using "when" would not be correct at all.
2. Homework is done between lessons. An assignment is a larger project, which may need several days or weeks to complete. It certainly can be done at home.
3. No, it's not exactly the same. "Having saved" = the result of saving money.
4. The phrase is "to be sick of (something)". "Sick over" is not correct in this sentence.