What is "bionn" - is it a form of 'Bi'? If so, what tense is it?

"bíonn cotadh air" - I assume "Is embrarassment on him" - "He is self-conscious"

Present habitual. "Tá cotadh air" (if it's even used) would mean "He's embarrassed [at this moment]", but doesn't say anything about what he's like generally.

Yep - it's the habitual present of bí:
Tá cotadh air - he is embarrassed/shy/self-conscious (at the moment)
Bíonn cotadh air - he is embarrassed/shy/self-conscious (habitually)

Bí is the only verb retaining such a distinction. All other verbs, the usual present tense (-(e)ann or -(a)íonn) does for both.

A Abigail, I know that's true of stative verbs like meas or teastaigh, but I thought the situation was slightly different in the case of action verbs. That is, ithim feoil capaill is closer in meaning to Bím ag ithe feola capaill than to Táim ag ithe feola capaill. Nach amhlaidh atá?

That's a bit like asking whether "sé" is closer in meaning to "he" or "it", and then observing that, well, it's used more often for inanimate objects...

I'd say it retains both senses but is more often used in the habitual sense - these days anyway.

Más amhlaidh inniu níorbh amhlaidh riamh. Is leasc linne a mhaireanns faoi anáil an Bhéarla brí ar bith eile a shamhlú le "ithim" seachas "I eat", abairt a bhfuil ciall amháin ag baint léi sa mBéarla, is í sin ciall na haimsire gnáthlaithrí. Dá mba sean-Ghaeil muid nó Géarmáiniseoirí dúchasacha ní rachadh an claonadh sin i bhfeidhm orainn mar a théann.

Deineadh an bhearna a líonadh nuair a d'athraíodh go Béarla ar dtús le "does be" agus "do be" ach bheadh aiféaltas ar fhormhór na ndaoine na "nuálaíochta" úd a úsáid.