Interestingly, "buí" is cognate, through PIE word *badyo- ("yellow, brown"), to Latin "badius", whence also words for horse color like English "bay", French "bai", Italian "baio". Also, "dearg" comes from PIE *dʰerg- ("to dim, to darken"), which gave English "dark" or Tocharian A "tärkär", which means "cloud". God, etymology is SO GREAT!
This reconstruction of 'bui' seems a little cheesy to me. In principle, Celtic languages have lost their initial b / bʰ sounds, and moreover the Italo-Celtic branch of PIE is known to have transformed (in some dialects) the labio-velar gʷ / gʷʰ to pure labials. Furthermore, pure b in Proto-Indo-European was very rear, thus, I won't dismiss the possibility of "buí" / "badius" coming from a proto-root with gʷ instead of b.
It's the generally accepted reconstruction. Sometimes also given as badios in Mallory's Encylopedia (badyo- coming from American Heritage Dictionary, badios coming from Indogermanisches etymologisches Wörterbuch; not sure which is considered more reliable. I would lean towards the latter). But, it's worth noting, that the word is only attested in Latin and Irish, so it could be a loan word into Italo-Celtic.
Yes. It's a feature of Irish phonology that extends to all words. I think it goes as follows: a schwa-like vowel is inserted into every pair of constants that has one r, l, m, or n. Corrections welcome.
(This also affects Hibero-English, which is why Irish people say things like "fillum" for film and "Dubbelin" for Dublin)