The sentence "I will never give up forever." is grammatically correct. Giving up forever means that the person won't ever try again. Never doing that means that the person will always try again.
The sense of the phrase never give up forever is that although the person might take a break from whatever they are attempting, they are stating that they will always return to try again.
If the intent was to apply forever to never give up, you need some punctuation between up and forever, such as
- ... will never give up. Forever.
This now has the sense that not only will the speaker try again, they won't even stop trying.
The second part of your question is:
I frequently hear "I will never ever…," which would imply "I" won't forever…, and I also want to know if it differs from "I will never forever…."
Will never ever uses ever as an intensifier. It conveys the same idea as the plain will never, but does so more emphatically. The three phrases are quite different.
Consider the following. Alice and Bob are names often used in the field of data protection (encryption, cryptosystems, etc). Suppose Bob sent Alice a message in a state that allowed a third party to read its plain text.
- Alice will never (ever) hold that against Bob.
- Alice won't hold that against Bob forever.
- Alice will never hold that against Bob forever.
The first says that Alice won't hold it against Bob today, tomorrow, or any time in the future, and that this stems from some aspect of Alice - if Bob did it again, Alice would not hold that against Bob, either.
The second says that Alice might hold it against Bob today, maybe even tomorrow, but at some time she will stop holding it against him. However, this is a one-off statement. It says nothing about how Alice would react if Bob did it again.
The third is similar to the second, with Bob in trouble for some finite period, but like the first, it carries the sense that something about Alice will eventually extend forgiveness to Bob each time.