Do I need to update to ios15 before switching to new phone

No, you can restore a lower version backup to a higher version. See→Transfer data from your previous iOS or iPadOS device to your new iPhone, iPad, or iPod touch - Apple Support

And it is perfectly alright to do this; there is no need to start over and reinstall everything, and it has the advantage that it will restore all of your app data. Starting over will mean a lot of app data will be lost.

No.

I also advise you to try not to use Migration Assistant or recover from a backup. A clean install is always good for perfomance and battery life. Try installing all your apps again as new, copy your personal data to iCloud (or another service of your choice) and/or sync with a computer (using iTunes/Finder).

This will help you stay away from debris like old files and settings that could cause you some trouble, you know.

If you don't care about this, then a computer backup is the way to do this Backup methods for iPhone, iPad, and iPod touch - Apple Support. It makes a more complete backup than the one using iCloud.

And no, you don't need to update to make a backup for a new device.