I am a resident alien of the U.S. by the substantial presence test. I fill federal Form 1040.
My spouse is a not resident alien filling form 1040NR-EZ separately.

1. Can I claim standard deduction on my Form 1040 while my spouse claims itemized deduction (state taxes) on her 1040NR-EZ?

If you are a resident of the U.S. then you can file joint return as residents even when your spouse has never been to the U.S. or is nonresident. Read Your U.S. Tax Return: Filing Status for Married

Or you can file separate return as resident and your spouse can file as nonresident. Yes, you can claim standard deduction.