'余計な話は要らない' and '事実だけをお願いします' are completely correct translations, however, I think if someone says either, they don't hide their annoyance any more after standing pointless explanation or in a emergency. Perhaps strict people might say that from the beginning, but I haven't heard someone say so as far as I can remember. People in a hurry might announce '時間があまりありませんので、手短に or 簡潔にお願いします' in a conference.
I would simply specify what I'd like to know:
I would say like this in case the person I've asked what has happened or other things start blah blah blah:
If it didn't work, I would try to find another person. It is just my way, so I hope other Japanese people give their opinions. :D
As for the last question, I would say '勝手な想像に過ぎませんが', and in a casual way '単なる想像だけど'.