1. Déarfaidh muíntir Uíbh Faoláin í, ag maíomh as an gcuma ’na bhfuaradar tosach ar an dá mhuíntir eile.

Tosach a dh'fháil - I don't think this means the same as "tosach a bheith agam - to be in the lead". Does it mean "to get one over on someone"?

2. bhí sí imithe amach doras a bhí in íochtar an tseómra - the original text said "in iachtar". This should mean "lower". Can it mean "the door on the other side of the room"?

3. Níor thug Gormfhlaith ná Sitric ná Amhlaoibh fios ná eólas do ar an uisce-fé-thalamh a bhí ar siúl acu i gcoinnibh Bhriain agus Mhurchadh. Ní fhéadfaidís é. Do scoiltfeadh air nó do leogfadh sé amach é.

I inserted a "t" in scoiltfeadh - it had said "do scoilfeadh air". According to the dictionary this should mean "he would burst" or something like that. What does it mean in context? He wouldn't be able to keep it in? He couldn't keep it to himself?

4. Díoghailt - meaning "revenge". How is this word pronounced? Is díoghlaim pronounced exactly like díolaim? Are they etymologically the same word in fact originally?

1. Gain precedence over
2. No idea, perhaps.
3. I'm not sure that t is correct: Could it not be scaoilfeadh - he would let it slip? Certainly the sense is that could not be trusted with the information.
4. pass

However Ó Dónaill has "Ba dhóigh leat go scoiltfeadh air" he looked fit to burst, but also under 2. Part "Ná scoilt do bhéal air" "don't open your lips about it"

I am thinking Aonghus that scoiltfeadh would be pronounced scoilthfeadh in Cork Irish. ie the lt is pronounced lh along the same lines as fáilte pronounced fáilthe, and as the -feadh is pronounced -headh anyway, the "t" doesn't add anything in this word and may have been missed out for that reason...

Tharlódh an ceart a bheith agat. Tá an ciall soiléir pé scéal é.

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Tosach a dh'fháil - I don't think this means the same as "tosach a bheith agam - to be in the lead". Does it mean "to get one over on someone"?

To get ahead of, to gain precedence over

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2. bhí sí imithe amach doras a bhí in íochtar an tseómra - the original text said "in iachtar". This should mean "lower". Can it mean "the door on the other side of the room"?

The lower part of the room. I think in the old houses they talked of "uachtar an tí" where the gable end would be and the fire place and "íochtar an tí" would be the opposite end of the house. Amhlaoibh Ó Luínse speaks of all this but I'm not entirely sure what "íochtar an tseomra" is to be honest with you now I see Ó Dónaill gives "íochtar an tí" as downstairs or the ground floor. You would hear people say "Tair aníos dtí'n dtine" which would suggest that wherever the fire is is the "upper" part of the house.

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3. Níor thug Gormfhlaith ná Sitric ná Amhlaoibh fios ná eólas do ar an uisce-fé-thalamh a bhí ar siúl acu i gcoinnibh Bhriain agus Mhurchadh. Ní fhéadfaidís é. Do scoiltfeadh air nó do leogfadh sé amach é.

I inserted a "t" in scoiltfeadh - it had said "do scoilfeadh air". According to the dictionary this should mean "he would burst" or something like that. What does it mean in context? He wouldn't be able to keep it in? He couldn't keep it to himself?

Yes, literally "he would burst", he would let the proverbial cat out of the bag. He wouldn't be able to keep his trap shut. Scoilt + ar with/without a subject pronoun is a common enough construction:

ba dhóigh leat air go mba bith beo éigin an pochán úd nuair a scoilteann air faoi theas na gréine (AC:79)

Raghaidh sé [madra] i ngleic leat agus le cách eile chun go scoiltfidh sé air féin = until he bursts with food (AC:56)

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4. Díoghailt - meaning "revenge". How is this word pronounced? Is díoghlaim pronounced exactly like díolaim? Are they etymologically the same word in fact originally?

/d'iːlt'/ I presume. And yes, presumably, díoghlaim would be díolaim. They seem to be just archaic spellings. Are they etymologically the same word? Yes, I would strongly think so but you would need to check DIL first to be sure.

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I am thinking Aonghus that scoiltfeadh would be pronounced scoilthfeadh in Cork Irish. ie the lt is pronounced lh along the same lines as fáilte pronounced fáilthe, and as the -feadh is pronounced -headh anyway, the "t" doesn't add anything in this word and may have been missed out for that reason...

I'm told that in northern Corca Dhuibhne, -lth- prevails but in the south of the peninsula, more or less south of Trá an Fhíona, -lt-, is or was the norm.

As for "do scoiltfeadh air" consider other forms lacking an obvious subject pronoun:

chuaigh agam an obair a dhéanamh
chuaigh dom an obair a dhéanamh
Dh'éirigh ar an ngaoith
Dh'éirigh liom an obair a chuir i gcrích

Well tá orm rud a dhéanamh lacks a personal pronoun. In fact, I have failed (do theip orm? or do theip sé orm?) to keep a good grip on which impersonal constructions have a pronoun, and which do not...

Shroich leis an obair a dhéanamh - another example from Ó Dónaill

Theip orm - no "sé". "Loic sé orm" = he failed me, he let me down. Nowadays, though, you'll hear "do lig sé síos me" as a calque of the English.