How long does it take to hear something back after dropping my recruiter packet? What is the likelihood I can get my preferred assignment?

First off are you a SGT?
Secondly, you will hear something in as soon as a week, but I'd say closer to 3 weeks. Once you submit your recruiting packet, they then have to send the request to your branch. Your branch could always deny you based off of numbers. However, if your branch allows the request, you will then be sent an ATTRS email with a school date.
For your second question, if you volunteer, you are more likely to get where you want to go, but it is still based off of positions available. A lot of the assignments in USAREC are based on gender, rank and skin color. I wanted Texas as my first choice. I got denied, but I got my second option of Florida. Most people that volunteered, got within their top 3.

As of last week they CG and CSM were putting out that they are over strength so it may take a while. They went back to individual mission to see who they are going to keep and who they are going to get rid of. It will also depend on if you are active or AGR. Both sides are over strength.

When I went through I had very little say whatsoever in where I went but if you want any of those places it's highly likely as they have a terrible reputation. They were recently ordered to stop calling Mid-Atlantic the Mid-Titanic. Let that sink in for a minute. I'm sure they'd take anyone motivated enough to go there.

I'm going to say be careful what you wish for.