Is "still" being used correctly in "I love you still"?

Yes, this is appropriate usage. As you noted, "still" is an adverb, so there's no confusion that it might be modifying "you" rather than "love".

It's basically used this way to create a grouping effect and add emphasis. In "I still love you," the assumed grouping is still love you. In this version, the grouping is love you still, equivalent to still love you. This emphasizes even more than it's his particular love for her that is ongoing, rather than his general capacity to love.

Yes, I love you still is standard, though you’re right that still most often appears before the verb.

It’s hard to say why the lyric is like this. Likely just for the rhyme or rhythm of it. The whole chorus just barely holds together, grammatically, poetically, or musically ...which is totally appropriate, given the situation, and a large part of the song’s charm.

If you want another theory, though, I’ve got one. ;)

Often people arrange sentences to put an element they want to emphasize last. See what I did there? It works! Adverbs that usually go before the verb are especially sweet candidates for that, since it is standard to put them at the end, but unusual enough to attract notice.

And behold, I am with you always, until the end of the age. (Mt 28:20)

So I think what’s being emphasized here is the word still.

I love you, still. Is grammatically correct.

You're probably used to

I still love you

Simply because it sounds better, and is correct as well.