1. Ann commanded/ordered that he (should) go there.
2. Ann suggested/proposed that he should join the club.
3. Ann insisted that he (should) wash the car.
4. Ann requested that he (should) go see a doctor.
5. Ann decided/determined that he (should) go to Chicago with her.
6. Ann desired that he (should) come back early.
(Are they all grammatical? Can we omit 'that' in each sentence?)
Leaving "that" in is fine.
Omit "should" from all of the sentences.
Yes, all of the sentences you provided are grammatically correct. In each sentence, there is an embedded clause expressing someone's request, suggestion, command, or desire. The word "that" can be omitted in each of these sentences without changing the meaning. Here are the revised versions without "that":
1. Ann commanded/ordered him (should) go there.
2. Ann suggested/proposed he should join the club.
3. Ann insisted he (should) wash the car.
4. Ann requested he (should) go see a doctor.
5. Ann decided/determined he (should) go to Chicago with her.
6. Ann desired he (should) come back early.
In formal writing, including "that" tends to be more common, but in spoken or less formal language, "that" can often be omitted.