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: 
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!
*“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*?”**