Of course he can't get the case transferred to a different state. He broke California law and is being prosecuted by the State of California. How exactly could the an Illinois prosecutor, judge, and jury handle that? He is going to have to travel back to California for all court dates.
There is an very small chance that the JAG at his base might try to cut a deal with California to have the military handle the case under the UCMJ. For something like a DUI, I doubt that will happen. In fact, he could end up getting NJP for this in addition to the civilian charge.
Yes, he could get kicked out. If he was ordered into alcohol treatment by the Navy, then this would probably be considered failing to complete the treatment.
As for how long separation takes, it varies by case. With only a year left on his enlistment, they might just deny his re-enlistment and have him ride out this enlistment rather than going through the trouble of involuntary discharge procedures. Or not.
Whether or not the government pays to move your stuff when he separates will depend on the characterization of his discharge.
If he hasn't already he needs to:
Get a lawyer in California for the DUI case
Notify his command (including his security manager) that he has been arrested and has a pending civilian criminal case
Contact his local Navy Defense Service Office in the event that his command wants to initiate any UCMJ or discharge proceedings against him.