Can a course count for both a degree requirement and a core curriculum requirement?

I’m majoring in chemistry so would chemistry and physics courses I take for my degree requirement also count for my natural science and technology requirement?

Absolutely. Here is the Chemistry checklist.

https://cns.utexas.edu/images/CNS/Deans_Office/Student_Records/Web.DegPlans/2016-18_Checklists/BSA_CH_1618.pdf

Notice on the left that the Science and Technology core requirements are marked as being fulfilled by major requirements.