This showed up as a fill-in-the-blank with the choices "deos" and "dei." Obviously, "deos" was intended, but choosing "dei" results in a different sentence which also seems to be me to be good Latin and means "We gods love." Right? Doesn't either answer produce a decent Latin sentence? In which case this exercise needs to be changed somehow.
You've missed the point. Latin allows nouns in the nominative case to be added in apposition as a kind of co-subject in cases where the verb already indicates a first or second person subject. So "Dei amamus" is a plausible Latin sentence where the subject is both "we" and "gods." In that case, WHAT or WHO is being loved is left unsaid, but the sentence works as Latin. If you already know that "gods" is supposed to be the object, then, yes, you have to use "deos," but the sentence as it was presented to me asked me simply to choose a word to complete the sentence, and with both "dei" and "deos" as options, there was not a single correct answer.
As an example, check out the first clause of St Patrick's Confession (originally written in Latin):
"Ego Patricius peccator rusticissimus et minimus omnium fidelium et contemptibilissimus apud plurimos patrem habui Calpornium diaconum filium quendam Potiti presbyteri . . ."
The verb here is "habui." The subject is everything from "Ego" up to "plurimos" and includes a pronoun ("Ego"), a proper noun ("Patricius"), and a long noun phrase ("peccator . . . plurimos").