Do I need new SIM card for new phone

Probably not, but check with your carrier.

Some carriers can and will switch your plan electronically (this happens when you migrate your content), some expect you to switch the SIM yourself (the migration will usually tell you, or your carrier will), and some few will issue a new SIM.

This varies, as some carriers will have device-change fees.

iPhone devices have had compatible SIMs for a while, so the swap is feasible on any recent model.

Remove or switch the SIM card in your iPhone or iPad - Apple Support

Learn which size SIM card your iPhone or iPad uses - Apple Support

Not unless you change wireless carriers.