My wife set up an LLC in 2023 (single member). No business activity in 2023. How should we file federal and state (North Carolina) tax return (if needed)?

The activity of the LLC is reported on your joint return. Since there was no activity, there is nothing to report. 

Thank you. Can I assume that she doesn't need to report for both IRS AND North Carolina tax return?

Is the tax deadline for her LLC is March 15 or April 15? I know for some cases it is March 15.

Correct 

Thank you. It's good to know that she doesn't need to report for both federal and state tax return, so in this case my understanding is that the deadline for filing is April 15 (not March 15). Please correct me if I am wrong.

The North Carolina Annual Report costs $200 each year for LLCs.

https://www.llcuniversity.com/north-carolina-llc/annual-report/

Yes. The LLC income and expenses are reported on Sch C as part of your personal income tax return.