When trading in my old iPhone do I remove the SIM card from the device before mailing it back to Apple?

You don’t HAVE to remove the nanoSIM, but it’s certainly a good idea to do so, and it only takes a few seconds.

However, you ABSOLUTELY DO want to “wipe” your phone of all personal data by following this procedure:

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

I personally always do but I don’t think it’s required

It doesn't matter. Remove it if you want to keep it.