In the sentence: I like Diet Caffeine Free Cherry Vanilla Coke.

I is the subject. Like is the verb.

Are the words Diet Caffeine Free Cherry Vanilla adjectives for Coke? Or is that whole phrase a noun since it is a thing?
It is a noun phrase in any case. Diet, caffein-free and cherry-vanilla (or cherry and vanilla) are adjectives modifying Coke. The Eiffel Tower is a single thing, but Eiffel is still an adjective modifying tower.
Students: We have free audio pronunciation exercises.
It's a little off topic, but have you ever heard of Diet Caffein-Free Cherry-Vanilla Coke Mr Micawber?

And one question: I guess I can use cherry-vanilla and cherry and vanilla keeping the very same meaning, can't I?

Frankly, I don't recall, YoHf, but I've heard of similar drinks-- Diet Pepsi Lime, Fanta Funky Orange, etc. Yes, I just hyphenated the flavours to make the noun phrase more intelligible. There's even an internet acronynm: DCFCVCWL stands for Diet Caffeine Free Cherry Vanilla Coke With Lemon.

The Coca-Cola website lists these products:

Major Products of The Coca-Cola Company in North America

Top Five Brands
Coca-Cola classic
Diet Coke/Coke light
caffeine free Diet Coke

caffeine free
Coca-Cola classic
Carver's Original Ginger Ale
Cherry Coke
Chippewa Water
Coca-Cola C2
Coca-Cola classic
Coca-Cola with Lime
Coca-Cola Zero
Dasani flavored waters
Diet Barq's
Diet Cherry Coke
Diet Coke with Splenda®
Diet Mello Yello'
Diet Nestea
Diet Northern Neck
Diet Rockstar
Diet Sprite Zero
Diet Vanilla Coke
Full Throttle
Mello Yello
Minute Maid juices and juice drinks
Minute Maid Lemonade
Minute Maid Light Lemonades
Minute Maid Light
Pibb X-tra
Powerade Option
Red Flash
Seagrams Ginger Ale & Mixers
Sprite Remix
Vanilla Coke