Can you catch tonsillitis from your dentist?

Unlikely. Not likely unless there is mucosal surface contact... And even then i would say the chances are still low. While tonsillitis is a bacterial infection and is by definition contagious.. It's not like a viral infection that is easily spread via mucosal surface contact and through the air and expelled droplets and/or inanimate objects (door knobs, keyboards, etc.).
Not likely. If your dentist or hygienist had tonsillitis while treating you, it is unlikely that you would contract it from him or her. The dental professional is required to wear gloves and masks, as well as provide infection control barriers to prevent cross contamination.
Don't think so. Anyone can pass germs to someone else, we use masks and gloves to prevent transmission of disease.