No, the R is not omitted in any variant of English. We all say the R in 'naturally' : even those of us with non-rhotic accents (eg people from England and Australia) pronounce the R in this word. Given that you're a Chinese speaker, I suspect that you have difficulty hearing the T and R combination (often blended to a 'chr' sound) when it's followed closely by an L.

No it isn't skipped. The "r" sound is in there. https://www.youtube.com/watch?v=a_YR4dKArgo

No. First, this is regional and dependent upon the accent but I believe that there are no accents in which the "r" is not pronounced, it might just be pronounced differently. The bigger change between accents in English is if one pronounces the "tu": I say "natcher-ally" where a British English speaker might say "natch-rally." If you want to hear pronunciations better, checkout www.youglish.com. You can actually hear the differences between US, UK, and Australian accents. R sounds in any language are often the most difficult to reproduce. If you're having trouble with it, don't worry. Just keep trying.

Nope. 🙂