Could it be that these two characters are not compatible with each other?

What is "petty theft"? Why did he suddenly mention this?

In legal terms, "petty theft" refers to the act of stealing property or belongings that do not have a high value. The boy's mother is feeling a bit down, so he tries to cheer up her up by reminding her she has a Ph.D, until he realizes that the name of the certificate is not hers - indicating that she stole it. Ex: The suspect was arrested on suspicions of petty theft. Ex: The man was charged with petty theft after he stole a radio.

What's "put down" mean?

"Put down" is a phrasal verb! Here it means to criticize someone. It can also mean to stop holding something and place it on a surface, record something in writing, euthanize or kill an animal, write down or enter someone's name for a competition, or lay a baby to sleep. Ex: I've put your name down for the competition. Ex: We had to put our dog down when he got too sick. Ex: We just put Lilly down to sleep. Ex: My lecturer was determined to put me down when I messed up the project.

Is "carwash" a place?

A "car wash" or "carwash", both spellings are acceptable, is a place where you go to clean the outside and, in some cases, the interior of vehicles. A "carwash" can be self-service (where you wash the car yourself), fully automated (where a machine washes the car) , or full-service with attendants who wash the vehicle. Also note that it may also be used to refer to an event where people pay to have their cars washed by volunteers as a method to raise money for some purpose. Ex: I need to take my car to the carwash. Ex: My brother works at the carwash.

What does "so bad" mean here?

"So bad" in this context means "a lot" or "so much". "So bad" is informal and idiomatic, however, it is very common in everyday conversation. Here's a couple examples: Ex: I want the job so bad. Ex: I haven't seen my family in forever. I miss them so bad!

Could it be that these two characters are not compatible with each other?

Yes, it seems like these two people have a rather uncomfortable relationship with each other, based on the interactions between them in this video.