Was what these cops did illegal? in texas!?

Stop whining obey the laws in place.

There is no way a cop can prove at trial that the vehicle was going one mile over. Radar is not that accurate.

Your daddy must be a decent chap to have numerous warrants.

If he was driving without a license, he's lucky he only got arrested and got the car towed.  In some jurisdictions, you'd get shot.

It's just too bad your mom didn't get arrested, and that you didn't get arrested either.

Pretty good chance they stopped him because they knew he had a warrant. They just made up that excuse so he wasn't agitated while they confirmed the warrant. Being deceptive about the reason for the stop is legal, as is towing the car. I can't count the amount of times someone said they could be there in 5 minutes and an hour later they don't show. 

Is what illegal?  Arresting a man driving without a license?  No, it's not illegal to arrest a person driving without a license, even in Texas.  If you are one mile over the limit, you are over the limit.  The law doesn't say, "50, but you can drive 5 miles (or any other number) over and it's okay."  Your mother had the option of driving the car away.  What's not planned?  Towing cars which for LEGAL purposes cannot be driven by the owners?  Okay, I'll tell you.  The situation wasn't planned.  Maybe Daddy should get a license.  You have a license.  Maybe you should drive his a$$ around.

He did not have a license, she did not have a license, and he had Warrants which are good until proven false in a court of law.  So there is the car there, It is towed away to impound.

YOU are irrelevant.  All legally done. It is in IMPOUND.

They can do that if they are denied permission to search the car without a warrant. They will tow the car to impound and get the warrant and search it legally then.

It's probably perfectly legal. You need legal advise. You won't get it here. Consult a lawyer or a public defender. 

Some cops get kickbacks from the tow companies.  But, sometimes, if the car is in a dangerous, busy location - they just want to get it cleared as quick as possible.