Is there still kart racing?
Also are 1 and 3 on steam aswell?
No and No, but if you really want to play the Ty Trilogy on your PC you could always try emulating them. From personal experience they emulate nearly perfectly.
does he have to eat from only one side of his face or does he have another little mouth on the other
For some reason Ty's mouth goes from the one side of his head to the other, however when you bite in the game his mouth works normally:

It's a weird design choice.
and the third game had bits where you drive around lol (though maybe the second one did too? i think those were just races though)
Both the second and third games had an open world where you would drive in. The one in the third game is just a mess and the one in the second game could've been designed better.