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: 
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)
*“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*?”**