Does deleting an esim erase my information, e.g., my phone's name?

Deleting the eSim just removes your phone number and service plan. It has no effect on your other data or information whatsoever.

Also, if the iPhone is locked to a carrier, then it will not unlock it. Only the carrier it is locked to, can unlock it.

If what is locked is the eSIM, and is requesting a pin, then removing the eSIM will stop the pin requests. This, again, does not remove any of your info.

To remove your information and details, and get it ready for someone else to use, you must completely erase the iPhone, not just the eSim.

Follow the steps in the link below.

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

Once again, this will not unlock the iPhone if it’s locked to a specific carrier. It will still be locked to that carrier, and will require a SIM card from that carrier to get cellular service.

Of the iPhone is locked to a carrier, only the carrier it is locked to can unlock it. No other carrier can unlock it, nor can Apple unlock it either.

The carrier lock, and the SIM card lock are 2 different things.

Deleting an eSIM will not delete most of your information - just information related to your cellular plan. But even. If you fully reset and erase the phone, if it is carrier-locked, that lock will remain. Only the carrier can release a carrier lock.

If your phone says it is still locked, then it is still locked and removing the eSim won't unlock it. You need to work with the carrier directly as they are the only one who can fully remove their lock.

I read that "deleting" the esim (I believe that is what I have) will "unlock" it so that the phone can be sold and reused.

What you read was not correct. The data on your phone has been erased, but the phone is still locked to the original carrier, as you will see if you try to set up the phone to use another carrier.

As it stands now, if you sell the phone, you will be selling a carrier locked phone.