PhD study for staff?
If there is a gap in your knowledge, and there is a course being offered that is a good fit for you, you could either enroll in the course or audit the course.  One course per semester maximum, please, so that it is clear where your priorities are.

One PhD is enough.  Math and computer science are close enough.

Yes, your self-confidence will grow as you build up a publication record.

Congrats on getting the job!