Need to trim 1/2" for new carpet. Doors are yellow pine from 1903 and very hard. Do I want to use a blade with very fine teeth to minimise chipping and splitting?

Use a 40 tooth carbide blade in a circular saw. Most of what you will be cutting will be a rip cut and you don't want too fine a blade. Use a straight edge as a guide to get the best cut. Place masking or duct tape either on the door where the saw shoe will be contacting door or on the bottom of the saw shoe to avoid marking up the finish.

1/2" is a bit more than I usually do but I use a power planer, no chipping. If you must saw score your line with a sharp blade and then apply masking tape to limit the splitting and scuff marks on the surface.

Id use the skill saw and as ballangerb1 says just score it, as for the planner, I've never had much luck on the tops and bottoms of doors due to the tear out when the planner leaves the edge of the door. If you do use a planner instead just add a piece of wood to the bottom edge of the door to prevent tear out

Thanks for all the help!