Can you get strep throat even if you do not have tonsils?

Not Uncommon. Although tonsillectomy greatly reduces the frequency of strep throat, it is not uncommon to still have such an infection in chidren.Thus you still should be concerned if your child is sick with a sore throat.
Yes. A tonsillectomy removes the prominent palatine tonsils on either side of the back of the throat. However, there is tonsil like tissue located in patches around the throat and roof of the throat. That and the rest of the throat tissue can become inflamed with a strep infection. Removal can reduce the intensity of symptoms but does not eliminate the risk of strep.
Highly unlikely. Highly unlikely. I've had a patient who developed not only tonsillitis but an abscess after a previous tonsillectomy. It happens if residual tonsil tissue remains after the tonsillectomy.