Legal Question in Family Law in California

Yes, the courts regularly make "interlocutory" orders for support, and most give priority to hearings on such orders. You say "going through a divorce," but not what that means. If the divorce proceeding is already on file, you should file an "Order to Show Cause" re: spousal and child support, as well as to establish a temporary custody schedule. If the divorce is not yet on file, you should file for the OSC at the same time you file for divorce, or at the same time you file your response if your spouse files the petition. Of course this all presumes you have less income and assets than your spouse. If you make more, you'd be the one paying in most cases.

That is possible. You can file an Order to Show Cause for temporary spousal and child support. Support is based on many things, including the parties income and child support takes the timeshare into account. Go to https://www.cse.ca.gov/ChildSupport/cse/guidelineCalculator

to calculate what you should be receiving for support prior to filing a motion.