I think that they are all valid. Although there is a difference in the tense of each paired clause, I believe that each sentence has a legitimate use. If I were concerned about any of the sentences, it would be #3.

Yes. To put it simply, you have two ideas in your sentences. One is the phrase "under the impression" and "to be very thin." When using was it means it is now in the past. This can mean there is a change for the idea. When using is the idea continues to be true. Here is what I mean: I was under the impression = The impression changed because you no longer believe so. I am under the impression = I continue to believe so. She is very thin = She continues to be thin She was very thin = She has now changed and might have gained weight. There are other grammatical ways to show changes in English. All your sentences are using changes in a not so clear way because "is" only shows continuation and "was" shows the finish in the past. Let me know if that is clear for you. Enjoy your studies.

The first sentence is the correct one. You begin in past tense and end in past tense. Having said that, you might think that the second sentence would also be correct, but we don't really say it like that. The most common way to say that saying is like the first sentence. Hope this helps!