I agree that "here's ten dollars" is the most common way to say it, at least among most Americans I know.

But if someone said,

"Hey, Scotcher, here are ten bucks. Go buy us a case of the cheapest beer you can find."

the "are" не режет ухо. You'd blip over the "are" without noticing it.
Totally disagree again. If someone said "hey Scotcher, here are ten bucks... " I would hear it as clearly as if they got the words in the wrong order. I wouldn't care, but I most certainly would hear it.

Unless they were presenting me with male deer or rabbits, which are countable nouns, in which case "here are ten bucks" would be grammatically correct.

That's a sign that the plural version is still part of the living language, not just an artifact of antiquated standards.
No-one said anything about the plural being antiquated. What we said was that dollars/ bucks in that context is a non-countable noun, therefore it doesn't take the plural at all. You are talking about an amount of something, not a number of things.