The answer to your question depends on what you mean by R. The sound /r/ is not a vowel and neither is /ɹ/, which is the R-like sound present in most English accents. However, in Northern American English for example, there is also the R-colored schwa /ɚ/. This is a vowel, or at least some phoneticians consider it one. I'm not a phonetician and I can't honestly say I know why some sounds are considered consonants and other vowels, so I can't make this any clearer.
As for the third formant and formants in general, I don't know what your acoustics knowledge is. I will assume no knowledge here.
When you pluck a string of a guitar, it starts vibrating at many frequencies at the same time. These frequencies are called partials and a sound with more than one partial is called complex. Most sounds you hear are complex. In particular, the speech sounds are always complex with many partials. It turns out that a human brain uses only a few of them to distinguish between speech sounds. These partials are called formants. The third formant of a sound it the third lowest.
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