How do I get a private car seller to refund me back my money for a car that failed smog?

He said he he would refund you, so take the refund. I don’t think you have a good case for the repair costs since you voluntarily did those, but...

Can I sue the city?

You can, but it’s not likely a successful suit. You hit a visible and stationary object, which is generally considered the driver’s fault.

Car Insurance claim?

Yes, assuming you don’t have loan agreement that says you must repair the car.

Other than saying to make the appropriate objections and answer truthfully, this question is unanswerable without a thorough review of the facts.

Will my insurance kick in if I'm found to be at fault in court? Or will I have to pay out of pocket in that case?

To answer the question you asked, your insurance company is obligated to provide you with a defense attorney if you are sued.

How can I get out of a contract when services rendered were very poor?

Just cancel and tell them they breached the contract. Failure to provide services in a reasonably acceptable manner is a ground for breaking a...

How long takes for. PI lawyer pay you for the compensation once they teceived the check from the insurance company?

One week is nothing. Talk to your attorney who is getting paid to address your issues.

What can the BOD do?

You have provided zero context of the situation. We have no idea what the rights of the parties are. We have no idea where you stand in all this....

Can I get the California DMV to waive the registration fee if the previous owner didn't pay the fee?

No you can’t. This is why you verify current registration when you buy from a private party, because registration follows the car not the owner.

I bought a car(Mini) from a dealership in cash used with lower miles? I had for a few hours right away there were things wrong

Very possible. A review of the purchase agreement, repair records, and ancillary facts would be necessary. Call a lemon law attorney, any of us...