May a person be an Enrolled Agent (EA) and a Certified Public Accountant (CPA) at the same time?

You could but after you become a CPA it wouldn't help to be an enrolled agent. You need experience as well as passing the test to be a CPA so you might want to be an enrolled agent while you wait.

You can be both although it is unusual and there is little point in being both unless you have tax clients in several states. Getting your CPA is harder but has more flexibility. If you passed the SEE you would have the EA designation and representation rights while you worked on your CPA requirements. This makes you more valuable than someone who is just a staff accountant. If you just want to do tax work, being an EA is fine.

yes but that would not make sense unless you like taking tests.

A CPA can do what an EA can do. But an EA cannot do what a CPA can do.