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)
We use ellipsis a lot in spoken English. It's certainly not good written style though!

I might have a conversation like this with friends:

Alright? Coming to the pub tonight?

Naa (No). Too tired. Went out last night.

Stay out late, did ya (you)?

Yeah. Got home about 2 in the morning. Wife wasn't happy!