iPhone 7's 3D touch features and Touch ID not working after repair

You have two (potentially) separate issues here:

1. You may have damaged the Home Button during the repair. The flex cable part of the HB is very fragile, as is the flex that runs underneath the LCD heat-shield. That would cause issues with the TouchID and click functions, depending on what kind of damage there is.
2. The 3D Touch feature could just be a poor quality replacement screen.

Start by putting the original screen back to see if the 3D Touch issue is resolved. If it is, then you probably have a cheap or defective replacement screen. As for the HB, do a thorough inspection, preferably under magnification, to see if there is any damage to the HB or it’s flex cable and connector. The HB cannot (usually) be repaired but if the damage if to the screen flex, a new screen would resolve the problem.