Can I move my family and household after I set everything at my duty station?

You can move your family after, but it's going to be very expensive paying for two separate households. You're only going to receive BAH for your duty station and you won't be able to stay in the barracks while receiving BAH.
There's no reason your wife can't move unless she's in legal trouble, you don't need to be physically present for the case, and flying back for court will be cheaper than renting a place for months on end. If she is in legal trouble, she can still move with the courts permission, even if she's on probation or parole. Your spouse can be seen by PT once you're at your new duty station.

This is a complex question with many factors, but the bottom line is that you should work with your chain of command to get your family moved at the time that is best for them. As a former commander, I always looked out for the best interests of my Marines while still maintaining mission readiness and capability. Regardless of where your wife had her accident, the lawyers can continue to work for her even at your new duty station. She can also continue to get the PT she needs. I suffered a traumatic brain injury in a car accident nearly two years ago and I am still in the process of settling the case and I live 200 miles from where the accident happened. My lawyers are working on the case and I am getting the medical care I need in my community. Work with your chain of command and let them work for your family.

Private, you can move your family any time that is convenient for you and your wife. If you are able to find and afford temporary housing and continue to pay for your family's housing, then you should not be faced with any normal issues arising from maintaining two "residences."