"We saw the dogs running."
Translation:Chonaiceamar na madraí ag rith.
Is "chonaiceamar" the past tense of "feicimid"? Or is it a different verb?
Yes, chonaiceamar is the past tense of feicimid. The past tense forms of feic seem to have their origin in a different verb, akin to “went” vs. “go” in English.
"Chonaic muid" is fine and would be more common in the northern half of the country.
Can you just put “running” mext to “dogs” the way you do in English? Having learned this construction in English as a foreign language learner I still tend to “translate” it in my mind as “the dogs that ran”, because that’s the construction that most resembles the natural way of saying it in my first language (even if the relative clause gives a different meaning to the English sentence!) So. Does the Irish sentence work exactly like English? Because if I had to guess I would say something like “na madraí a raibh ag rith”—and if this sounds unnatural or too long or unnecessarily wordy or whatever, does the grammar at least make sense? Does it have a different meaning?
I'm not sure that I understand your question. "We saw the dogs running" - the dogs were running, they were barking, they were rolling in the mud. I know that the dogs were running because we saw them running. Chonaiceamar na madraí ag rith.
na madraí a bhí ag rith - "the dogs that were running" - we saw some dogs running, we saw some other dogs swimming in the lake. "the dogs that were running" didn't go into the lake - ní dheachaigh na madraí a bhí ag rith isteach sa loch.