In California, his parental right can't be terminated just because he was deported. It is not easy for one parent to get the other parent's rights...

Can I appeal a child custody order?

You should be able to appeal, but time is of the essence. Consult a family attorney in your area asap.

Will I get arrested if I don't show up to my restraining order hearing

Unless you are in the hospital, go to court. You should consult with a criminal attorney. The attorney may be able to appear on your behalf.

If a woman lies says she's on birth control. Does the guy have recourse if she gets prego?

Child support is for the child. If you don't want a child, don't engage in baby making behavior.

My wife took the kids to a different country on a 3 week vacation and decided to stay there keeping the kids.

I agree with Attorney Alexander. You should hire an attorney ASAP and begin the process.

Can a women with a tempary restraining order be arrested for liying in family law for getting a restanning order aginst a man

It's technically possible but may not occur. Cover your bases. Consult an attorney.

Moved to NorCal from SoCal. Ex blocking visitation due to age of child and objections on her flying in a plane.

I agree with the other attorneys. You also may want to consult your 9 year old daughter and see how she feels about it.

Seems like the lease you both signed is valid. If you continue to be bullied, see an attorney. You have the "key", literally and figuratively.

If I claim my child on my tax return, do I owe my child's mother half of the return?

You cannot claim your wife as a dependent. If there is no Court Order, you don't have to give her half the tax return now, but the court might...

If I leave my house when I file for divorce - do I abandon my rights to the property of my house?

No, where you live should not determine the nature of the property you own.