Are the sentences natural?

The first two sentences are fine. Since you are asking what is "natural," no, i don't think that the third sentence would naturally suggest itself to me.

Yes, although I could suggest "Jack is a three-times-in-a-row winner" as a replacement for the last sentence, as many native speakers (not I) would avoid the word "consecutive" for fear of appearing prententious or because it is not part of their everyday vocabulary.

In British English, performing the same achievement three times consecutively is often called a "hat-trick", a phrase originating in the game of cricket.