faulty digitizer for sure.
A screen replacement with an original quality screen will solve. Unfortunately many of the replacement screens are low quality--especially the earlier screens, and the repair community has learned a lot about best practices for screen installation in order to prevent digitizer ghosting.
Why not give it a DIY try? You can get a quality original screen for $50-$100 with the ic/home flex soldered in already so it will be a plug and play install. Avoid cheap screens.
yes, that is the right part.
I would use iFixit guides, and just know that for the iPad mini it is really really important to FIRST disconnect the iPad mini battery before you pull the LCD screen.
My personal tips:
1.) Use a heat gun and start at bottom right corner moving up. Use a thin tool like an iSesamo opener (totally worth it to have)
2.) Once you remove the 4 LCD screws, know that the LCD is hard to pull up without cracking it---make sure your tool (i use the black plastic spudgers) is under the LCD, but not under the metal shield below it. go slowly and carefully.
3.) It is okay to cut through the two pieces of tape at the bottom of the LCD
4.) after unscrewing the shields remember--disconnect battery first!
5.) Before assembly--use a sticker remover pen to get all traces of old adhesive off the screen.
6.) go over the frame with alcohol to clean any remaining residue
7.) Apply a set of ipad mini adhesive stickers to the frame before assembly.
8.) Assemble in reverse and remember to pull the cover from the inside of your new digitizer--i pull the outer cover off too so i can see dust better as i'm sealing it up
9.) pick off the camera bracket on the screen---it causes too many last second screen breaks and is not worth the risk.
10.) Work with some canned air to keep dust free while sealing.
11.) press all around with microfiber cloth and add a little heat for that perfect professional finish....it will look brand new.