What you are hearing is a "flapped" r sound (which does sound like a D) followed by AN, but the AN is reduced to A', so you get "ar a'". You sometimes hear a flapped R in "upper crust" British speech in, for example, "mirror" which sounds like [midda] as opposed to standard American, which sound like [meer] to my British ears.
I understand that, to me it sounded like eren. However, Irish is set up to flow really well, so when spoken it can sounds like the words join together. Its very different from English in that way where we like to hear the start and end of every word. Obviously, different regions with their own pronunciations complicate things more for learner's
Ar an Máirt = On Tuesdays. Máirt in singular in the same way that Tuesday is singular in the dialectal English expression "I walk of a Tuesday".
"On Tuesdays" is the right interpretation because Irish present tenses are "habitual" in essence. If you want to be specific about a particular Tuesday, I suppose you could use the continuous, "Tá mé ag siúl Dé Mairt" or "Beidh mé ag siúl Dé Máirt".
No. It should be accepted, and, in fact seems to be the preferred translation for ar an Máirt (Meaning ar an is generally habitual, Dé ____ for specific one)
I walk on Tuesday, can be habitual: Monday I do the washing, go down the pub on Friday, Saturday watch the rugby .... you do not have to always use a plural day to imply an habitual. I eat fish on Friday...seems unfair to insist on a plural Tuesdays, when both the singular and the plural are possible. Grant both, perhaps.