iPhone screen becomes unresponsive and taps random things

The digitizer is what senses touches and is most likely bad. The digitizer is built into the screen, so a screen replacement will be required. Make sure this is done only at the Apple Store or Authorized Apple service center.

I don't know about forcing you to update, but you may want to state that you don't want to update as any tech may do it as a courtesy not knowing that you do not intend to update. At that time, they will be able to tell you if the service requires an update.

If you ever wanted to upgrade your phone, now would be the time. The cost of the screen replacement is $129 and there may be deals with your carrier and a trade-in option that would make a newer phone very reasonably priced or even free if you consider the cost to repair your current phone. In addition with battery health typically lasting only about 5 years, you would also be looking at a battery replacement in the near future. I am not in the sales business, that is just how I would look at it if it was me.