Translation:The children are playing in the garden with the dog.
I originally thought the same. In the States it would be interchangeable, but yes, of course the Irish sentence is the other way around in word order. It all means the same thing anyway. If the children where playing in the garden with the dog and messing up Mom's tomato plants, they would for sure be in deep trouble!
I also have a yard and a garden, but I would think of the garden as the garden beds, so specifically non-grass plants like flowers, bushes, and trees, while the yard is the flat/uncovered/undecorated dirt or grass. Similar to a lawn, but to me, a lawn sounds neater, as though for display purposes, while a yard is more casual, possibly fenced, just a general open area. I'm from Australia, so it's interesting to see the difference between countries!
Perhaps US "yard" is possible as a translation?
Yard https://www.focloir.ie/en/dictionary/ei/yard#yard__5 (US) garden: clós, gairdín, garraí; next to house: clós, sráid, front yard: clós tosaigh.
Backyard https://www.focloir.ie/en/dictionary/ei/Back+yard 1 in brick, concrete etc: clós, clós cúil, cúlchlós, cúlghairdín; 2 (US) garden: cúl an tí
No, the present continuous is formed by 'tá' + subject + ag + the relevant verbal noun. Verbal nouns come in many forms and you have to learn them on a case-by-case basis.
Here are a few examples to give you an idea of the variety of forms: imeacht (going); labhairt (speaking); déanamh (doing); tógáil (building); fáil (getting); ithe (eating); ól (drinking); siúl (walking), roghnú (choosing).
Your example is in the present tense, but the Irish is in the present progressive tense. "The children play in the garden with the dog" would be Imríonn na páistí sa ghairdín leis an madra. Irish differentiates between "they are playing (at the moment)" and "they play (in general)", not too dissimilar from English.
I can't hear a difference between broad dh and broad gh. C.f. "dhá" audio at https://www.teanglann.ie/en/fuaim/dh%c3%a1. And Wiktionary transcribes the Connacht versions as /ɣɑː/ and /ˈɣɑːɾˠdʲiːnʲ/, respectively. So both are shown with the voiced velar fricative https://en.m.wiktionary.org/wiki/%C9%A3.
Is there no difference between broad 'dh' and broad 'gh'?