Hiya!
No. The Drow has no need to wiggle his finger or utter any words in order to levitate. It's an ability. The Sorcerer class is explicitly an intuitive understand of how to cast arcane spells, with a metamagic twist.
Not all innate magic is the same. Not all innate magic = spells.
Semantics determined by individual DM's I'm thinking. For my, "if it's not able to be done using the natural reality of the current Prime Material Plane ('campaign world'), then it's magic". Being able to "tap into" or "understand and manipulate" is irrelevant ...the end result is "it's magic".
Sure. I agree with that. The problem is that Counterspell does not counter magic. It counters spells.
Or not. There was a rule in there for the DM to ignore that, but with advice that it could unbalance things. Not that psionics are relevant here. We're talking counterspell and nothing allowed a 3e wizard to counterspell psionic levitate, even though counterspelling existed in 3e.
Now
I'm going to play the semantics card...
Page201, PHB:
"What is a Spell?
A spell is a discrete magical effect, a single shaping of the magical energies that suffuse the multiverse into a specific, limited expression."
To me, this is defining of what a 'spell', mechanically, is in the 5e game. Now, most spells in the PHB are going to be what are commonly assumed when the word "Spell" is used (re: Fireball, Dispel Magic, Gate, Polymorph, etc), to be sure. BUT...what I just quoted does NOT make that distinction; it's defining 'spell' as an expression of magic...like a Beholders eye powers.
Just to be clear, the issue here isn't really what is "magic" or not. It's about what is specifically a spell, for the purposes of the spell counterspell or abilities like the Mage Slayer feat that operate specifically on "spells".
As I semantically pointed out... spells aren't spells only.

Unless, of course, the DM rules they are. Which is cool. As I've said, I have no problem with a DM ruling that way and I'd happily play in a game where that was the case.
And, before it goes there...it probably has already, but anyway... I
DO NOT think this needs any sort of "errata". Just leave it as is and let individual DM's and groups decide how far a "spell" covers in terms of what is/isn't, well, "magic". It makes for a MUCH more interesting and diverse playing field. Differing opinions, outlooks, preferences, etc when running D&D is a GOOD THING! It should be lauded, not vilified. Well, imnsho, of course.
Even under the current rules you can't Counterspell a beholder ray, even though they are magic. The Beholder is a perfect example under the original rules of a creature with magical actions that are like spells, but are not spells and therefore not counterable. So far I've not heard about anyone who has been confused or complained about it being inconsistent.
See above.
In my game, yeah, you can
Counterspell a Beholder's eye stalk effects... otherwise the DM would have to say "No, you can't
Dispel Magic on your buddy who you believe is
Charmed from the Beholders first eyestalk...because it's not magic".
^_^
Paul L. Ming