happy girl; It is a document stating that you carry no grudge towards the other party
of the broken engagement. Some people will not date the girl or boy without it.
Some people don’t mind as long as their research bears satisfying results.
The side that did the breaking usually in this case would require a shtar,
Though both sides might want one for shidduch purposes.
(To prove the other side has no resentment and (evileye) towards them.)
In the case of a mutual breakup, both sides might definitely want one,
why not?
In a case like you mentioned, R”L one should try to beseech the other side
till they are willing to forgive.
I’m not clear as to you your last question.