If I have genital herpes, could I be immune to oral herpes?

No. All forms of herpes are caused by a virus that lies latent in the body until it decides to cause external lesions. Once you have the virus it remains in the body and the strength of your immune system plays a role in whether you will exhibit lesions or not. Having one kind does not keep you from having another and actually makes you more likely to have breakouts in multiple areas.
No. No, they are two different strains of the same virus, getting one does not mean you are immune to the other, and getting one does not mean you have the other.
Yes. There are several types of herpes viruses. Classically type 1 causes oral herpes or cold sores and type 2 causes genital herpes. With the sexual habits of today, type 1 could also cause genital herpes. A blood test to see if you have antibodies to type 1 or 2 can be done. Many people have the antibodies, but have never had an outbreak that they are aware of.