I am not either a native speaker or teacher.
But I know about the question you asked.
You can't omit the 'to' in this sentence : "I got Tina to send the mails for me."
The follwing are for helping you understand the matter easily.
I made/had/let Tina send the mails for me . This is correct.
I made/had/let Tina to send the mails for me. This is not correct.
You must omit the 'to' in the upper sentence.
Each 'to' in the upper sentence must be omitted.
The ommittable 'to' is in the follwing sentences.
I bade him to guard the house.
I will help you to pass the exam.
Unfortunately, I have no idea about daily conversation.
In school grammar you can not omit the 'to' in the sentence you mentioned.
Student or Learner