You're welcome, glad to hear it. Hopefully they also have a handle on stock for you 👍

That looks like a Glacier Bay Diverter Cartridge, Home Depot sell replacements under the Danco brand.

Those cartridges are usually just press fit in there but you might be able to just take that extension piece to the store and match it up if you don't feel comfortable pulling the cartridge out.

The Valve Socket is required to remove the diverter properly and then you would be able to match it up correctly yes. You can use a crescent wrench on the socket but I usually use a long screwdriver through the whole in the socket for extra leverage.

When you remove the valve check to see if the washer/oring was left behind in the valve body itself and if there is any damage inside so as to not ruin your new valve.

Definitely get a set of Shower Valve Sockets and take off your Diverter valve to get a replacement. They are usually under $10 and you can get a handle set to match or change up the whole look.

This video applies to most diverter valve like yours: https://youtu.be/KjmAZjfpEq8

No more chance of a problem, definitely the best idea until it's fixed as well 👍

Besides water bill, if you get a clog the water will overflow from your bowl (happened to one of my residents)

I would suggest replacing your fill valve, most of the time your old valve overfills the tank because the diaphragm inside has gone bad. They are cheap and easy to replace plus added peace of mind.