Even though I know what "forever and ever" means I would like to know if I can write a love letter to my girlfriend and say something like "I'll love you forever and ever" or even "my name and her name forever and ever" any advice will be appreciated.
Yes, you could use that phrasing, with a couple of reservations.
1) If you really want to make an impression, don't use the contraction I'll, especially in writing: it's a bit informal. Write I will in full.
2) According to Ngram, in British English, we say exclusively "for ever and ever". In American English, "for ever and ever" is the most common, but "forever and ever" does also occur.
So, I recommend that you write:
I will love you for ever and ever