Would you feel awkward asking a coworker for a ride to the auto mechanic to get your car after work?

Of course not.  I've done it myself many times.

Not if I knew them well and you can trust that person if you don’t ask you won’t receive help hopefully the person will have the attitude to treat others as they would want to be treated. 

No, he/she might be very happy to do that for you and might even need a favour from you one day.

No, i wouldn't.  Especially if it was someone i talk with at work and if it was on their way home.  

I suppose you can always take Uber if all else fails.

Not at all. Unless it involved an hour in traffic across town. My friends (co-workers) and I often do this kind of stuff for each other. 

It would depend on how well I know the coworker.  Most coworkers have at least a few "coworker friends" that they talk to daily at work - it wouldn't be strange to ask one of them.  

But, if you barely say "hello" to a coworker - then it would be awkward to ask that coworker for a ride.  

wait min. if why you need some body take you to the auto mechanic to get your car to be pick up after work? why you can just take the bus or taxi or have your family to take you there instead asking some body who work for the same job place as you. that like not nice  if was a friend of yours is same thing .your co worker has own things to take care of and plus she or he might thinking why you cant get a ride from your parents or take a bus . alot of co workers doesnt like to take adventage from other people .

No, but I would hope that I do know my coworkers well enough not to ask someone it would be especially inconvenient for so I don't put anyone on the spot.  It's not like you're trying to mooch a free ride into work every day - it's a one off.  I'd be more than happy to help anyone out if they asked me and work on the presumption that most people you you talk to every day at work are the same.