Are professors required to perform research as a necessity to teaching? In my country, in order to teach mathematics (or any subject) to college students one doesn't have to have a PhD. Just a good academic record in the masters level of one's subject and one must clear a (reasonably difficult) national level teachers eligibility test in one's subject. Having a PhD and research publications is a bonus but not mandatory.

So my question then is - would it be necessary to do research once you have secured a permanent faculty position? By necessary I mean, will I face any difficulties in teaching or explaining concepts and ideas to students if I don't spend some time doing research, publishing papers, learning and asking new questions in a given field?
Over the course of a long career in teaching, the nature of your instruction will necessarily need to change as both our scientific understanding of the world and our society change.  This is especially true for undergraduate and graduate instruction, where the knowledge is more specialized and changes more quickly, but I would argue it is also true for much of one's earlier schooling as well.

To be a good instructor, then, I think that one absolutely must keep up with the evolving nature of one's field.  But does that require doing research?  I don't think so (except, of course, for advising graduate students who are required to do research).  Remaining involved in research is definitely one of the easiest ways to keep up with a field, but it's not a panacea either: research is typically extremely focused and narrow, and being up to date in one's particular research sub-field does not necessarily mean you will learn about everything relevant to instructing your students in a course that is only related in the more general sense (e.g., a platinum electrocatalyst researcher teaching introductory organic chemistry).  One can consume information about research without producing it oneself, however, and many other forms of professional education are also available.

Bottom line: doing research is a great way of keeping up with your field, but not necessarily sufficient and not necessarily the only way.