Is it True that FP and Finn Break up?

Yes, they broke up in Frost & Fire. Finn tried to get back to Princess Bubblegum in the following episode, Too Old, but he rejected him because she was focusing on some lemon child (I forgot his name). At the end of Too Old, Finn berkata he would try to fix things up with Flame.
In Earth & Water, FP becomes Flame Queen. When Finn asked if they were still going out, she berkata no but they could still be good friends if he was completely honest to her.

Yup, but in the Earth and api special they become friends again but not boyfriend and girlfriend.

Ya, but like with Tony and Jessie on Jessie, I don't feel angry bout their breakup. I feel at peace with it.

Not really, in that episode FP berkata she needed some time alone but in the following episodes she berkata they can still be friends and she took over as her her father

no