Are these questions natural?
In this video: https://www.youtube.com/watch?v=PXhDlYREJyo at 0:26, the subtitles read "Do you have Irish". Is that structure used only in Ireland?
I also remember hearing an American person who is not uneducated say "Do you know English" to an Italian person.
The simple way to express this idea is "I speak English", or "I know English". Since knowing something is equivalent to having that knowledge you could say, for example, "I have a little Irish".
It would sound a little strange for a person to say "I have a little English." when speaking in English. It would be better to stick to the specific verbs "speak" or "know". So while "I have English" is possible, it is not the recommended way of expressing this concept.
Question 1: No, "Do you have English" is not correct. It could however be argued that it's part of Hiberno-English and thus acceptable. Either way, learning to say this isn't advisable.
Question 2: One expects this structure to be exclusive to Ireland, where it is commonly used (especially in Western regions).
Why? As Ireland transitioned from being Irish-speaking to English-speaking, certain words, phrases etc. were naturally carried over, but also syntax.
The standard way of asking this question in Irish is:
An bhfuil Bearla agat?
The literal translation is:
"Is there English on you?"
However, because there isn't a word in Irish equivalent to "have"*, this same syntax is used and this could equally be translated as:
Do you have English?
This persisted and is still in use today along with many other examples that sound perfectly natural to an Irish person but strange to anyone else.
*Or several other obviously useful things like "Yes", "No", or "Thanks"
As a native English speaker (but not a language professional)
"Do you know English?"
This would be a perfectly reasonable way to ask someone if they know a language. "Do you speak English?" is far more common though.
"Do you have English?"
This is just weird, a language is something you know not something you have.
Googling the phrase reveals a bunch of results but most of them are the sequence of words as part of a larger sentence where "English" is acting as an adjective.
