"Questions like this" or "questions like these"

It’s correct. This refers to the particular question; questions means other questions which are similar in nature to it. They do not have to agree in number.

There could be many questions that are like this one question.

Suppose I am in a clothing store looking at shirts. Nine are blue and one is green. I point to the green one and ask, “Do you have any other shirts that are the same color as this one?” There is only one green shirt under discussion, so I must refer to it in the singular. But the store might well have many other green shirts – there’s certainly no reason to assume they have only one other green shirt. So “other shirts” is appropriately plural.

If I was looking at nine blue shirts and two green shirts, then I would ask, “Do you have any other shirts that are the same color as these?” As there are two green shirts, I now refer to them using a plural.

If I had previously seen another green shirt, and I was specifically asking about that one, I might ask, “Where is the other shirt that is the same color as these?”

All that said, in context, saying, “questions like this one” means pretty much the same thing as “a question like this one”. In the first case we’re talking about all possible questions that are “like this one”; in the second case we’re talking about a single, generic question that is “like this one”, without particularly implying that there might not be others that are also like this one.

But it would change the meaning to say “questions like these”. There is only one question presently under discussion, so to refer to it in the plural would arguably be just wrong. I suppose you could say that a plural could mean this question and others like it, but the whole point of the statement is to identify other questions as being like it. So if the writer had said “questions like these”, I think my immediate response would be to wonder, this one and which others?

You only asked one question and it is to that one question that this refers.

If you say “questions like this” then you are referring to one type of question. But “questions like these” means more like many type of questions.