Ford Focus LX LS (2006) - High Revs at Car Start Up

[dohzer]

To keep things brief:
1. I had problems starting the car; sometimes it would sound 'weak'. I suspected battery or alternator.
2. The car wouldn't start after approximately a week of non-use.
3. I took car in for service, they replaced the battery but I was always sceptical about whether they serviced the car since the first time I went to pick it up, they hadn't actually serviced it so I had to leave it there for another day.

4. When I got the car back I noticed it was revving up to ~3k RPM at start up which I remember sounding higher than normal.
5. Slowly it got worse to the point it was revving at ~4k RPM at start up.
6. The other day (after a week of short-trip use) I went for an hour long drive, and when I would set the cruise control at 60km/h, the car would slowly keep accelerating above the set point with zero pressure on the accelerator.
7. Finally the "Engine Systems Fault" light came on after about an hour of driving, and it started struggling and 'sputtering' a bit.

Any idea what would cause:
a. High revving at car start up (before it settles down to idle).
b. The accelerator to push past the cruise control set point?

Does this align with a standard problem/fault?