修车奇遇

I sent my car to an auto shop for repair. They gave me a phone call for diagnostics and cost estimation. When I picked up the car several days later, they claimed the car has been repaired and tested. But after I drove the car for 10 minutes, the same problem happened again. I had to send the car back to their shop. They came up with a new problem of the car and another cost estimation, which I had no choice but to approve. When I came again to pick up the car, they claimed again the car had been repaired and tested on a 15-minute drive. But after I drove the "repaired" car for 10 minutes, the same problem occurred again. I had to send it back again. They came up with another new problem and another cost estimation, which, again, I had no choice but to let them do. Finally after the 3rd repair, the car is running okay now; but I paid for the 3 repairs. 

so my questions are (1) on their service policy, they should have given a written estimation, but I have never received any written estimation. On all the phone estimations, they claimed they found "THE" problem. (2) do I have reason to suspect they mis-diagnosed for the 1st and 2nd attempts of repairing the car, because the identical problems happened within minutes of my driving? (3) For the 1st time and 2nd time I picked up the car, they should have scanned the code and found there were still problems, but the obviously didn't bother to do that , when they said they "tested" the car (4) For the 1st and 2nd repairs, they obviously lied when they said they tested the car, especially for the 2nd time when they said they had done a long 15-min test, because the exactly same problem happened within my 5-minute driving. 

Had the mechanics have any responsibilities (meaning wrong doing) during this process? Also, after I found their repair was unsuccessful the first time, could I have just walked away without paying them?