For a few years, my car occasionally gets 'stuck' at 3000 when pulling off, and I have to use the manual + button on the side of the gear shift to move up a gear; then it's fine. There have been no other issues, such as loss in performance, gear slipping, cranking sounds, or any warning lights, so I've just lived with it. For the last two weeks, there has been what I can only describe as a 'bassy' sound from the middle-ish area of the engine. Again, no performance loss or error lights, etc. Also, when I hold an idle rev at 1000rpm, there's a slight 'rattle' like sound. The only change to the car recently was the driver's side front and back tyres. One was reconditioned, and one was new.

Assume this is a wet clutch model - if not then most of the suggestions may be invalid (just trying to help).
The rattling may mean that it is a dry clutch unit which I know little about - sorry again.

As usual, suggest you start from the basics (and monitor at each step) which include:
What is the mileage and maintenance history of any oil and filter changes?
Change the filter (and oil, if not done for a couple of years), rip the filter apart to check for debris (plastic bits especially)
Use Forscan to check for codes and use all gearbox PIDs to compare measured and demanded values and that the signals from the lever are being registered.
Check and clean earth connectors at the front chassis rail (if Focus has them)
Check the wiring from transmission to PCM.