Translation:The dog is playing with the comb.
When is sugradh used?
Sugradh seems to indicate children at play
The post by Fear_na_mbróg at that link seems to differentiate between súgradh and imirt fairly well.
That post seems to suggest sugradh might be better here, unless the dog is playing some formal game with the comb.
It might well have been the better choice here, although the definitions in this dictionary entry suggest that the “Of pastime” meaning of imir might make the most sense for a dog without supercanine attributes, e.g. a dog who has adopted a particular comb as his own and likes to bat it around.
What's the difference between tá an madra ag imirt and imríonn an madra?
"Tá an madra ag imirt" is present progressive (it is happening right now), "Imríonn an madra" is like simple present, I guess.
It is, as far as I know. When I was a pupil we said "present continuous", my children learned "present progressive".
Irish and English differentiate between "plays"/imríonn and "is playing"/ag imirt. They are not interchangeable, even though some other languages use the same structure for both tenses.
Would "is at play" be OK (and a more literal translation that also happens to work in English)? I tried it out (out of interest) and it was not accepted but perhaps it had not been considered and therefore not included in the list of correct answers. It would give an intuitive feel for why "ag" appears in the Irish phrase and would provide insight into the logic of the grammar if it were correct but I don't know enough yet about conjugation of the verb denoting "to play" to judge whether it is a valid interpretation or just an interesting coincidence in this particular case.
It would be misleading to accept "at play", because you might then expect ag ithe to be translatable as "at eat", and ag rith as at run.
So, yes, in this specific case, it could give a more intuitive feel to the translation (except that the term "at play" would usually not involve the verb imir in Irish, with the noun forms ag sugradh or ag spraoi being more appropriate). But the phrase "at play" is a bit exceptional - most other verbs can't be used that way ("at work" and "at prayer" do work this way, but these are the exceptions, rather than the rule).
Present continuous is corner with him/bionn + ag + inf. Why here is ta????? Is it another tense???
I think you have your definitions mixed up. Bíonn is used in the "habitual present".
For the present continuous, you'll typically use the simple present tense Tá + pronoun + ag + "verbal noun" (not infinitive).
You can use the habitual present tense with the + ag + "verbal noun" structure too, but it certainly isn't required.