Well first off, you HAVE to have some kind of converter from mains AC to local DC. I'm not qualified to say if that conversion is easy to do if you roll your own: I do think though that dabbling in mains power without knowing exactly what you're doing is dangerous. I can say nothing other than you should get an off-the-shelf ac-dc converter. So you need to put that inside the box of the "unit" and just have a mains wire coming out of there.
I doubt if anyone here will encourage you to do the ac mains to dc 12v by any other means than off-the-shelf.
But you can get one that goes to 12V and then drop that to that 5 or 8 or whatever is required with components like 7805 which is inefficient but does work.
Go and have a look at Pololu
: they have a huge range of voltage converters so if you get the 12VDC sorted from the mains ac, they have various devices to get you to other dc voltages. I have their 2120 for example.
Then regarding the motors. It's not easy to control a dc motor as carefully as you might need. So you should probably investigate both stepper motors
to see if they can do what you need. Steppers as the name implies move in well controlled steps of usually 1.8 degrees (200 per revolution); might be good for the auger. Servos move to a degree position, usually between 0 and 180 and are good for raising and lowering.
There are libraries to drive steppers and servos. Have a look at Pololu for stepper driver modules which take care of the electronics. Servos are dead simple.
edit... And pololu have modules with h-bridges to drive dc motors. Steer clear of old technology modules that use 293 or 298 chips... very wasteful of energy. I have a Pololu 2130 and it's good.