What are the runway and taxiway separation standards?

Required separations between runway centerline and parallel taxiway centerline, and between taxiways and objects, vary by the critical aircraft's Approach Category (AAC), Airplane Design Group (ADG), and approach visibility minimums per FAA AC 150/5300-13B Appendix G. The runway-centerline-to-parallel-taxiway-centerline separation differs by AAC at the same ADG: for C/D/E-III it is 400 ft, while A/B-III is 300 ft (visual / not lower than 3/4 SM) and 350 ft (lower than 3/4 SM).

How far must a taxiway be from a runway centerline?

It depends on the Airplane Design Group and approach visibility minimums. AvPlot's Separation Matrix returns the exact runway-centerline-to-taxiway-centerline value, taxiway and taxilane object-free-area widths, and taxiway-centerline-to-object distances.