It is a synonym but I only ever hear it in religious Christian-based discussion. I would never use it in daily conversation. It generally means giving up a habit or object. If you are talking about the context in which you quit trying to do a difficult task, forsake would not work.

It's both formal and old-fashioned. It is rarely used. It means the same thing as 'give up' in certain contexts, for example if you 'forsake' alcohol for a period of time, then you have given it up. Or you can forsake/give up believing in something. But equally there are many situations where they are not synonymous.

'Forsake' doesn't exactly mean 'give up' in the same way... it's a bit hard for me to explain it. But really, I never use it and I never hear anyone else using it in their daily lives. It's a really old word and no one ever uses it.

Yes! It is a good synonym! ;)

I can't think why you'd be asked not to use phrasal verbs--except as an artificial requirement in some kind of classroom situation. "Give up" has more than one meaning. A word that means "to stop trying to make progress on a task" is "to abandon." "He abandoned his hopeless search for a way to turn lead into gold." You can also use "relinquished" here: "He relinquished his hopeless search for a way to turn lead into gold." A word that means "to admit defeat" is "surrender." "I surrender, we'll do what you want." "The American Civil War ended when General Lee surrendered to General Grant at Appomattox." To talk about letting go of something we own and giving it to someone else, we can use "yield" or "relinquish." (Here, "the floor" means "my turn to speak in a meeting.") "He yielded the floor to Senator Foghorn." "He relinquished the floor to Senator Foghorn." "Cars must yield the right of way to pedestrians in a crosswalk." I've _never_ heard the work "forsake" used except in the context of a breaking off a romantic relationship. In the traditional U.S. wedding service, each spouse speaks of "forsaking all others till death do us part." In a song, the singer might say something like "She forsook me and now I'm sad and lonely."