Thank you very much indeed, Jim. That's a great explanation. You cleared all my doubts. Forgive me for thinking out loud as I hope you can correct me if I am still making any mistakes.
Firstly, I didn't notice that it was (can only use "was", "is" not possible?) not reported speech! And universal falsehood. What a phrase! This is important because we are not talking about something that is always true. As a result, we can only use the past tense in sentences starting with "I thought it was". The "was" is not the result of backshifting. It is used instead to indicate that the believe was hold true in the past but is now found to be false.
Secondly, both present tense and past tense are possible in reported speech when talking about universal truths or things that still hold true at the present moment. This has been confusing me for a long time because I was taught to use the simple present in such cases and I thought *only* simple present is correct. It has been troubling me because sometimes I saw past tense being used and I always wondered why.
While I was waiting eagerly for your reply, I opened Randolph Quirk's A Comprehensive Grammar of the English Language, which I recently bought from Amazon. It doesn't mention the first point (or I just couldn't find it), but on the second point, it totally agrees with you!
You are really a great teacher!
Now, one last question. Please forgive my insatiable desire to get to every details. While both present tense and past tense are acceptable for the sample sentences you used, which of them sounds more natural? Are the ones inside the brackets less natural / common?
EDIT: I didn't see Barbara's post when I replied. Thank you very much Barbara. It's obvious that you also didn't see Jim's post when you replied. I am so glad to also get the confirmation from you. Once again, I can take your answer to the bank! I also want to thank Goodman, N2G and Marius for all your insights you have given me!