Are academic questions (used in textbooks etc.) owned by the publisher, or is anyone able to use them in their own content? I am wondering whether mathematical and scientific questions (or any type of question to be honest) is considered to be owned by the publisher? For example, if I see a math question in a textbook, am I allowed to use that question on my website, or is that legally considered an infringement?
To write this in more detail, the text of the questions is clearly owned by the copyright holder. The ideas embodied in the questions are not protected by copyright. It's usually considered plagiarism to completely rephrase and rewrite a question without citing the original source, but plagiarism generally isn't illegal. 

As noted in the downvoted answer, in some communities there is a broad acceptance of some kinds of copying when it comes to exercises, but I wouldn't want to endorse wholesale copying of homework exercises from one book to another. There are likely to be canonical problems that should be worked by every student in a particular discipline, and those would probably have the least claim to protection and the least interest in protection from the publisher. The more creativity going into the question, the more protection it has and the more enforcement you are likely to see.

Writing good exercises is challenging, but you should endeavor to do so for your website. You should cite to sources when you adapt an exercise from somewhere, and you should mention that an exercise is a classical example that all students should be able to do when you draw on something that Gauss or Euler proved as an exercise centuries ago.