FYI-- French is not the mother of English, but the stepmom, thanks to the Norman Conquest. English is part of the West Germanic language family. There are thousands of words in English of French origin, but that’s just an added layer of vocabulary.
I have only lived in America 71 years so perhaps I have missed something here. In my experience, Americans use house pet interchangeably with pet to refer an animal kept in the house (dogs, cats, etc.). For instance, I might have a dog who is a pet but is kept outside. That dog would be a pet but not a house pet. On the other hand the two indoor cats I have that drive me nuts daily are very definitely house pets. I do not ever recall seeing "Haus Haustier" used to refer to these pets and so I would think that either pet or house pet ought to be acceptable as a translation for Haustier. I'm guessing here that German does not differentiate where as English does. Might be wrong on that.....still learning.
Is there a grammatical reason why Haustier is neuter? I though with compound words in German that the gender is determined by the final element? Tier being the final element, isn't tier masculine? Or is this some weird exception and "Haustier" is more of an independent word than other compound words are?
You have to learn it when you learn the word.
i.e. don't learn "Löffel = spoon"; instead learn "der Löffel = the spoon" so that you remember that "Löffel" takes "der".
It's a little simpler in a compound noun like this one: compounds always take the gender from the last part. So if you know that it's das Tier (neuter), then you know that it must also be das Haustier. (And das Faultier "the sloth", das Trampeltier "the Bactrian camel", etc.)