Marriage and last name

[aalapsha]

I am a US citizen and I will be getting married in court very soon. My spouse is planning to change her last name. She is on an H1 status.
I was told that she would have to update her passport and social security with her last name. Will she also need to update her H1 through her employer before filing for a green card?

If she were to keep her maiden last name after marriage would that affect her green card application process (will they ask questions about that during the interview)?

Thanks.