This is what it says in the "Tips & notes" section about the is...mé forms (called the copula): "The copula is a special third form of the verb to be. It links the subject of a sentence with a subject complement, such as a noun or pronoun: for example, in sentences like I am a man, the woman is a cook, he is our friend, or that is a book. Therefore, bí is not used in these cases."
As far as I know it's similar to ser/estar usage in Spanish - is......e is used for something permanent (is fear e - he is a man). On the other hand ta se fuair ( he is cold) is something temporary and changeable. I might be wrong, but hope that helps. Also, i don't know how to do fadas ( the little accent above the vowels) on my ipad, so it looks weird when I type the Irish words without it.
It's not really a diphthong. Theoretically it should be one syllable, but sometimes it's more like two, like in this recording (sounds like "ta-im.") Technically in theory it's "tahm" (one syllable) with a slender m. There's many different accents in this language...I say it both ways but that's because I'm a beginner lol.
In some of my teach-yourself books, "I am" appears as "Ta'im" (sorry, I can't get a fada over that 'a' for some reason, hence the apostrophe standing in for it). In others, it gets presented as "Ta' me'" (again, the apostrophes stand in for fadas over the 'a' & 'e' -- sorry). Is one or the other form more common / widespread / current, or are they interchangeable?