1. I believe him to be sincere.
2. I believe him to go there.
3. I imagine him to be sincere.
4. I imagine him to go there.
5. I guess him to be sincere.
6. I guess him to go there.
7. I suppose him to be sincere.
8. I suppose him to go there.
9. I think him to be sincere.
10. I think him to go there.
11. I consider him to be sincere.
12. I consider him to go there.
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Are they all geamatical? Can we delete 'to be'?
2, 4, 6, 8, 10, and 12 are not correct. These infinitives all need to be "to be" infinitives, not action verb infinitives.
The other sentences are correct. I wouldn't delete "to be."