Sometimes I have seen people ask questions using sentences which don't start with an auxiliary verb:
You don't like to eat ice cream?
Instead of:
Do you like to eat ice cream?
Is it okay ask questions in both cases? I think its common around Internet people asking question like that but how correct it is?
Both are grammatically acceptable, but have different meanings. Saying "You don't...?" is a question you would ask to confirm what you already think. Saying "Do you ...?" is just a regular question.
For example, if you are thinking about buying some ice cream for dessert, you could ask your friend "Do you like ice cream?", and they would respond "yes" or "no".
If you're walking down the street and some is handing out free ice cream, and your friend doesn't take any, you might say "You don't like ice cream?" (you assume they didn't take any because they dislike it), and they might say "no I don't like it" or "I do, but I'm not hungry", or some other reason they might not want ice cream right now.
There is also the option of "Don't you like ice cream?", which is similar to "You don't like...?" except now you originally assumed they did like something, but they gave you reason to doubt your original assumption. Take the second scenario of free ice cream on the street again. If your friend doesn't take any you might say "Don't you like ice cream?" (because you assume everyone likes ice cream, but are confused because your friend didn't take any).
It is perfectly correct to say either. They both make sense and there is nothing wrong grammatically.
You would say "you don't like" at the start of the question normally if you already have a preconceived suspicion that they might not like ice cream but you are still not entirely sure. That is the only difference as far as I can tell.
"You don't like X?" is not a good way to ask a question.
If you reply, "Yes", then you are saying "Yes, I don't like X"
If you reply, "No", then you are saying "No, I don't like X"
The canonical example of why this is an inappropriate way to ask a question is
"Have you stopped beating your wife?"
