Will my HomePod automatically connect with new iPhone?

Your HomePod isn't paired to your phone, rather it is part of your HomeKit even if your iPhone XR set up your HomePod. Every eligable device signed in with your Apple ID will be able to access your devices in Home, including your new phone which you will no doubt restore from a backup of your retiring iPhone.

I have a couple of phones signed into my account and the second one, a retired 6S without a SIM, is used for diagnostics and on very rare occasions for other reasons. It behaves the same way as my iPhone 13; I can use it to operate any of my HomeKit devices.

When you set up your new iPhone and sign in to iCloud it will behave the same way as your XR.

I upgraded my XR to an iPhone 13 last week.

What I experienced was that my bedroom homepod had difficulties when I ask Siri about my calendar appointments.

My living room homepod was doing just fine. I resetted the bedroom homepod and now He is fine either

I can’t tell you the actual list of features but what you mentioned would be some; personal requests and phone calls. The wifi connection might also present a problem since it automatically connects to the wifi the iOS device (iPhone) you used to set HomePod up is on. HomeKit is fine and everything should load into the Home App on the new phone once iCloud is set up (using the same Apple ID)

Be sure to back up your old phone........before.....you set up the new phone. This way, you will have all the up to date data on the new phone when you restore it from the backup.

Everything in HomeKit will work. It uses you Apple ID. However it would be best if you did reset HomePod and set it back up with your new iPhone for certain features to work.