As an American, I use "real" and "really" interchangeable. The grammar is not what a professor would use, but it is common. For example: "That sandwich was really tasty" or "That pizza is real tasty." "The music is really loud!" or "That movie is real good." You don't need to use "real" in this way, but it is good if you recognize it when you hear someone else use it. Good luck!

It is common to use ‘real’ as an adverb, but it is still considered informal. Note that most uses of ‘really’ aren’t with its original adjectival meaning- it just means ‘very’ These aren’t real diamonds. I’m really tired. (No one doubts that I’m tired. ‘Really’ just means ‘very’. It has nothing to do with the meaning of ‘real’) I’m very tired. I’m real tired. (Informal, regional) She’s really engaged. (You doubted me and I’m confirming that it’s true) She’s real engaged. (No! We only use it to mean ‘very’)

It’s not correct, but very common for native speakers to drop the LY. I want it bad(ly) I’ll work quick(ly)

colloquially yes Americans do use "real" as an adverb sometimes like you have in your example sentence, BUT this is super informal and shouldn't be used in writing, it would only be used in speaking. Examples: I'm real excited about the upcoming movie release! This is going to be the best movie of the year! / Sometimes one of my friends is real mean to me and so i've decided not to talk to them anymore but again this is very US and very SOUTHERN US at that and would really only be used in speaking. It does make you sound more fluent, but it doesn't make you sound very intelligent. I would recommend branching out into other adverbs and intensifiers like "incredibly, insanely or extremely" for more variety in your speaking and writing.

In British English we never use 'real' in this sense. It's American English.