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.
These aren't correct in formal English. Here, the sentences say "Are you alright?", "Have you eaten anything", as you mentioned. The "Are"/"Have" are implied here, that's all. One finds this a lot in informal English -- words which can be guessed from context are sometimes dropped altogether. 

It's just a way to save time, similar to how contractions are used (though contractions may be used in formal English as well)