It depends on the dialect, and the function of the word. For example, in Connacht, it's /ə/ if used as a verbal noun, but it'd generally be /u:/ if used as the past autonomous (for first conjugation verbs), and /əx/ in the conditional.
In Donegal, it generally tends to /u:/ in all cases, I believe. I'm not sure about Munster.
No, she doesn't.
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Insofar as táim ag and tá mé ag are distinguishable at all, the vowel sound between m and g is the a of ag, not the é of mé.