That's how "bord" is pronounced in parts of Connacht. Other similar words don't follow the same pattern, but some of them feature an epenthetic vowel sound after the "r".
assuming it's not the slender r at the end that gives you pause, it could help you to notice the aoi sound in "faoin" and other words. So you can practice pronouncing faoin before chathaoir a couple of times if that's the part of word giving you trouble and then it should lock In. Then you may visualize hearing a word with that ee sound in the ending and knowing it is potentially a word that has aoi in the ending and using context to dial in on the particular word.
pronounce the following
cathair- cahaird, faoi - fee, faoin - feen, daoine - deenye, thaoir ( fake word)- heeird, cathaoir - caheeird,
-rd ending being how my ears tend to hear slender r
The fact that you've never heard it pronounced that way only tells us that you haven't much experience listening to spoken Irish, it doesn't tell us anything about whether it's a common pronunciation.
As explained in the comments that you didn't read before you posted your comment, this is the normal Connacht pronunciation of bord and would be familiar to most Irish speakers, even if they themselves don't speak Connacht Irish.
No. The proposition in this sentence is "le", which is usually translated as "with".
The proposition "ag" is used to say "has" - "Tá an chathaoir ag an mbord". (note the oder of "cathaoir" and "bord").
"le" is used to specify ownership, but only with the copula, not with the verb "bí" ("tá") - "is liomsa é" - "it is mine".