Grammatically correct but nuances are different. I pointed AT my cat and told them that IT IS a cat, but they didn't believe me. Using past tense can mean, that the cat is now no longer a cat. Perhaps, it is magical and turned into something else or that it is now dead. Same reasoning for the rest of the sentences... this person was a doctor --> this means that he may have retired, stripped of his licence etc. You should use present tense to indicate that he is still a doctor. Use present tense to indicate that it is still a perceived fact regardless whether the situation was in the past. In the old days, people believed that the earth IS flat. -- a fact, even though their thinking is wrong. In this case, you can also use past tense, ... WAS flat -- here you are telling that in the old days, people believed that it was flat but no longer now.

Yes, all the sentences are correct. Well done!

Correct, but all the sentences are the same though. In order not to have doubts, remember that when you retell what someone said you make one step back in tense: Present Indefinite becomes Past Indefinite, Present Continuous -> Past Continuous and so on. Will converts into Would.