Is haustier plural and singular?
Only singular. The plural is Haustiere.
Are there any other synonyms for "pet?"
Haus - house
Tier - animal
doesn't get much more straightforward than this :) You'll notice many German words are compunds like this.
Not really, no, unless maybe dialectally.
Wow penalised for a question mark what happened to typo
What exactly did you write? Usually Duolingo ignores punctuation (except unfortunately quotation marks as a certain other sentence indicates).