{o-hazukashii} no dakedo uchi no ie iro-iro wake atte... お恥ずかしいのだけどうちの家色々わけあって･･･ [It's] something {embarrassing}, but, in my [family] a lot stuff happened and...
iro-iro wakearu 色々訳ある There are various reasons [for something to occur]. A lot of stuff happen [that lead to other stuff happening].
{konotsuno ga haete-kitara mahou shoujo wo sagashite buchi-korogasanakucha ikenai} sadame rashii desu このツノが生えてきたら魔法少女を探してぶちころがさなくちゃいけないさだめらしいです It seems there's a rule [that] {as these horns sprouted out, if [I] don't search for magical girls and murderize them, that can't go}.
i.e. I must search for magical girls and murderize them.
sadame 定め A rule that's been stipulated. (from sadameru 定める, "to stipulate a rule.") This noun is qualified by a very long {relative clause}.
buchi ぶち To hit. To strike. (verb.) Emphasizes force. (auxiliary verb.)
korogasu 転がす To make something roll over. Trip over. To throw someone down. (not to be confused with korosu 殺す, which means "to kill.")
So buchi-korogasu means literally to throw down with force, which sounds practically like buchi-korosu, "to murder with force."
