I'm pretty picky about my neighbors too.
I have rental property next to me also. It has been empty for about 2 years and very quiet up until a month ago. The couple that moved in are 19 and 20. Loud, rude and just no morals. We listen to them scream and fight and cuss all day and ngiht. We are in the process of putting up an 8' picket fence. That will at least keep us from having to look at them anyway.
Dirt bikes are not street legal at all. If there are dirt trails anywhere around you, that is more than likely what they will be using them for. Hopefully you won't have any problems with them.
If your girl across the street is as bad as you say, she will probably love the new neighbors