Question: I'm having a hard time figuring out which is the correct form of asking this kind of question. I mean speaking strictly, this doesn't sound right: ***You alright?*** or ***You eaten anything?*** compared to ***Are you alright?*** and ***Have you eaten anything?***.  

So please enlighten me. Are those both forms correct or just something which is ignored for the sake slang speakers?

Answer: 
Those phrases are examples of ellipsis: the omission of words that can be understood from the context, or given contextual clues.

While ellipsis is not normally used in formal English, it is more used in spoken English, or informal English.
*“You alright?”* is not necessarily true according to the grammar rules. It is instead used for conversations with friends and relatives. The most correct sentence, in my opinion, would definitely be: **“*Are you alright*?”**