About how long will it take to find out if your packet was selected for 160th SOAR?

Depends on your qualifications and current need. I recommend contacting the recruiter for feedback rather than speculating. They will tell you if they have a slot and can give you a +/- 30 day (see what I did there?) answer. As stated, the recruiters may be backed up and have a good deal of review work to do. Packets go out to units for review after initial qualification screening as well. If qualified for a slot you then go into the chute for EGP and orders cut accordingly.

That depends...how much money did you put between the pages of the packet?

It really depends on how backed up the recruiters are. You may want to contact one of them directly.