Why did Elrond remain in Rivendell?
In the Hobbit and LOTR, the characters always visited Rivendell on the way to their deeds, or adventures. Furthermore, Elrond is the keeper of the Elven Ring, Vilya, which started in the Second Age during his and Gil-galad’s forming of the Last Alliance of the Second Age.
Rivendell was viewed as an enchanted sanctuary in the “extreme eastern margin of the wesern lands.”
Thus, after “escaping” to Elrond from imminent pursuit of evil, the hero can go face it in an entirely new direction.
Did Elrond ever fight in battle?
He has fought in the Second Age and beginning of the Third Age. However, I think it’s worth noting:
Did you wonder why would Elrond remain in Rivendell during most of the Third Age, or were you happy with what had happened?