This seems like an English construct translated directly into Irish. In Irish one would say "We brought the new dog home with us".
Thugamar an madra nua abhaile linn.
Bit ironic in an Irish language course that my Hiberno-English "we brought the..." is marked incorrect and the British "we took the..." given as correct. It is correct in Hiberno-English to say "bring to" AND "bring from". We would only use "took from" when a change of possession is implied, e.g. "he took the money from her".
As the first comment above points out, the "take"/"bring" overlap really comes about because you would usually use thugamar linn rather than thógamar in Irish, and tóg is usually only used for bring in the sense of "upbringing" (an áit ar tógadh mé - "the place where I was brought up/raised/reared")
No, because the adjective nua is qualifying the noun madra it means "new dog".
The English sentence "we built the dog a new home" is a bit tricky, because "the dog" isn't the object of the sentence, "a new home" is the object, and the Irish would be:
Thógamar baile nua don mhadra
I think in many cases Rinneamar teach nua don mhadra would be preferred, because it avoids any ambiguity in tóg, as a dog's house is often portable, and teach would be preferred even for a home for a person. While abhaile and sa bhaile are translated as "homewards" and "at home", baile on it's own is more likely to be interpreted as "town".
Thanks! I very much appreciate your thoughtful response! As you'll see from below, I didn't fully read the whole sentence — not really grasping the word order, OR the use of abhaile vs. teach! DOH! (BTW, your suggestion of using rinneamar to avoid the ambiguity of tóg is great!, too)
In Ireland, we use bring/brought in contexts where it is more grammatically correct (per Queen’s English) to use take/took. The correct verb depends on the the direction of movement and the perspective of the speaker. You won’t hear a well-educated Englishman say, “I’m bringing my better half to a party tonight.” He will take him/her.