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.
I typed the sentence (listening exercise) starting with “Tá mé…”, and it is not accepted, but there is no way to report that (no “My answer should be accepted” option) :)
If she said "táim" and you wrote "tá mé", then your answer shouldn't be accepted for a "Type what you hear" exercise. (And I say that even though I can't tell whether she's saying "táim" or "tá mé" and it doesn't make any difference, grammatically).
Since it sounds more like "tá mé" to me than "táim". I recorded the sentence and slowed it down. She does say "tá mé ag".
No, she doesn't.
Rather than record the audio that you hear and play it back, you can access this recording directly here. Your browser should let you play that back at half speed (Microsoft Edge does a much better job than Firefox for half speed playback), or you can download it and run it through whatever software you prefer.
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*.