ATT classic SIM. User travellling to India

Yes, but you would have to find a carrier in India that supports eSIM. The easier solution is to convert the AT&T nanoSIM to an eSIM, thus freeing up the SIM slot.

You can convert a physical SIM to an eSIM on the same iPhone, if your carrier supports it. Follow these steps:

1. On your iPhone, go to Settings > Cellular.
2. Tap Convert to eSIM.
3. If you don't see Convert to eSIM, your carrier doesn't support this option. Contact your carrier to move your phone number from your physical SIM to an eSIM using eSIM Carrier Activation or by scanning a QR code.
4. Tap Convert Cellular Plan.
5. 
6. Tap Convert to eSIM.
7. Wait for your eSIM to activate. Your previous SIM card is deactivated when the cellular plan on your iPhone activates.
8. Remove the physical SIM from your iPhone. Then restart your iPhone.

No. One physical SIM can be present in the phone at a time.

You could transfer your AT&T service to the built in eSIM and use a local physical SIM in India IF the phone is not carrier locked.

If you purchased the phone and financed it using AT&T's NEXT financing plan, it's locked to AT&T and you can not use any other carrier on it until you pay it off and get it unlocked.