Learn German: Er ist immer noch hier.

You are right that immer by itself means "always". However, when you combine it with noch to make the phrase immer noch, the meaning changes to "still" (as in "still happening"). It is a fixed phrase that you can memorize together.

Could I just say Er ist noch hier to mean "He is still here"?

Yes, absolutely! Er ist noch hier is perfectly correct and also means "He is still here." Adding immer to make immer noch just adds extra emphasis, like stressing "He is still here" because maybe you expected him to have left by now.

Why does immer noch come after the verb ist but before hier?

In simple German sentences, the verb (here, ist) must sit in the second position. Adverbs like immer noch generally come right after the verb and before words that describe a location, like hier (here).

Since this lesson is about adverbs of frequency, how would I use immer to say "He is always here"?

To say "He is always here," you just drop the word noch. The sentence then becomes Er ist immer hier, which uses the regular frequency meaning of immer.