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Only three words put E second and Y third, which is striking given how common the EY pair is elsewhere.
KEY, THEY, GREY, MONEY and HONEY all carry EY, but in every one of them the pair sits at the end. Pushing it to positions two and three leaves almost nothing standing.
That is why the count here is three rather than the dozens you might expect. The pair is common in English but almost never in this position.
KEYED is the only one in ordinary use, the past tense of key. FEYER is the comparative of fey, meaning strange or otherworldly, and GEYAN is a Scots word meaning considerable or rather.
On almost any puzzle, yes. The other two are rare enough that you would be unlucky to meet them.
Strange, otherworldly, or fated. FEYER is its comparative, though most writers would say more fey instead.
It is a Scots adverb meaning rather or considerably. Scots contributes a steady trickle of entries to five letter word lists.
Very little. They take three different openers and three different endings, so the pattern gives you nothing to narrow with.