Is the past tense for the word "earn" "earned" or "earnt", and does the word "earnt" even exist?
According to the Wiktionary, "earnt" is correct but not common:
This is an uncommon (<0.5% as common as earned in the British National Corpus) but entirely acceptable alternative form of the simple past and past participle earned. Still considered to be incorrect by many, who are largely unaware of the historical development of the English language.
According to the same Wictionary page,
Other verbs which can be conjugated in this way are: learn (learnt), dream (dreamt), spell (spelt).
"Earnt" clearly exists, both as a spelling and as a corresponding pronunciation that is distinct from "earned".
The formation of "earnt" is irregular, but not randomly so: the past-tense/past-participle marker takes or may take the form -t after /n/ in some other words, such as burnt, learnt, or after the phonologically similar sounds /l/ (felt, knelt) or /m/ (dreamt). The exact development of these kinds of irregular past-tense forms seems somewhat unclear (there was a separate question about it Why does the preterite of verbs such as "deal", "feel" and "dream" have a devoiced dental suffix?), but that doesn't change the fact that standard English includes many irregular past-tense/past-participle forms in -t.
Whether something "exists" in the sense of "is accepted as a word" is a matter of opinion. Sometimes, there is widespread agreement; sometimes, there is disagreement.
There is no simple principle that will allow you to objectively determine whether something "is a word". For example, "syllabus" is widely considered to be "a word" even though etymologists think that it originated as a mistake.
At most, you could determine whether some specific person or organization accepts the use of some particular word or word-form (such as "earnt").
"Earnt" seems to be uncommon, not only in American English but also in standard written British English.