You're confusing orthography and phonology. "may" is spelt in IPA as /mei/, so yes, it's a diphthong there that English represents using a vowel + Y for historical reasons. Also, there isn't a y sound in "mate" if you pronounce it at normal speed.
I don't understand what you mean by "by that reasoning". But there's no reason for the r in "beard" to have to be a vowel since it's followed by a consonant, since that's never stopped most other consonants before.
You're confusing orthography and phonology.
I know the difference. They always teach vowels and consonants as letters instead of as phonemes, and most people seem to use them that way, so I just have to talk about the phonemes corresponding to those letters. I also don't know IPA very well, and I can't assume anyone else does, so I tend to just say things like "y sound".
Also, there isn't a y sound in "mate" if you pronounce it at normal speed.
http://dictionary.reference.com/ has my as /meɪ/ and mate as /meɪt/. Vowels are a lot mor...
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