How do aircraft steer on ground? How do modern aircraft steer on the ground? Do they have a steering wheel which moves the front wheel (left/right).

I see that there is a Tug vehicle which pushes the aircraft backwards, so the aircraft does not have a reverse gear for sure.

Does the aircraft use the Engine thrust to move forward on the ground or does it have separate motor which propels the wheels?
All of these methods are used.

Nosewheel steering is common on most commercial aircraft using a tiller.  Rudder, differential braking, differential thrust and even motors in the wheels (experimental only as far as I know) are also used, as seen in this video.

Engine thrust is used to move forward but differential thrust might also be used when negotiating tight turns. (Details here)