Are you required to spend a certain amount of time at a gaining active duty unit before you are allowed to drop a SOF packet?

You cannot PCS straight to CAAS. It is not a duty assignment, it is a 9 day selection course.
There is no XRAY program for CA.

However, you do not need to wait a year to apply for CAAS. You can do so as soon as you are eligible on Active duty. That said, getting the physical done and getting a CAAS class date reserved will probably be a class 3-6 months after you get to your first unit.
Then, assuming you get selected, you will get a course date that is at least 3 months out, and maybe 6 months out.
You can get an exception to PCS to the course after 9 months on your first duty station, but odds are you wouldn't need it anyway.

Yeah, it's pretty standard (I think army policy) that you spend a year with that unit. You'd be hard pressed to find any commander that will sign a waiver for you on the basis that they just got you, releasing you within a year would be an unnecessary process for them when the slot could've just gone to someone who'd stick around in their unit.

Lots of required proceses! Once you figure out your transfer into the regular army and get into your first duty station, you could drop your CA packet shortly after. Remember, the packet itself is an admin cumbersome; time consuming. Add to it your A&S TDY which will not be immediately after dropping your packet. Positively speaking... you get selected. It will take a few months after completion of your A&S before the CA branch cut you order to Q Course. CA branch will not PCS until your time on station is met; or shortly after. Keep in mind you might not PCS if your unit is scheduled for deployment.