I’m looking for screen printing frames with a mesh count of 60T. I’d like to understand how to choose the exact mesh count on the basis of the characteristics of the design I need to print.

The choice of the mesh count does depend on the characteristics of the design, but it mainly depends on the type of ink being used.

With water-based inks the maximum suggested mesh count is 55T, you could go as high as 77T, but printing becomes more difficult and requires more experience.
With these mesh counts it's possible to print lines of a minimum thickness of 0.3 mm without problem.

To print an A4 (21x30 cm) design, you need a frame with external dimensions of 42x52 cm.

To print an A3 (30x42 cm) design, you need a frame with external dimensions of 47x68 cm.