The kitchen door is 34in x 81 3/4in. I learned quite quickly Lowe's and Home Depot don't carry anything near this size. I'm afraid to order a custom size online due to the extra $$$.

I think a couple of lumber supply's might carry 36x84. I could get that and cut it down, but then I have another problem - can I cut that much off a solid-core (not all-wood) door?

From what I understand most solid cores are filled with particleboard with wooden edges. If I cut too much and expose the particle core, can't I just nail/screw a new wood plank on for the edge? Should I just remove it's frame, install a couple 2x4s on the edges to compensate for that non-standard size, then use a standard pre-hung?

 

Most standard doors are 80 inched tall. The rough opening for that is 82 3/4 inch.
If you can remove the old frame and install a new pre-hung door without having to redo the wall over the doorway that would be the easiest solution. You could trim hollow core doors and install new frame in them but it is not as easy a job as it sounds. You have to be sure you have enough solid wood to hold the doorknobs on one side and hinges on the other. If it is truly particle board in the middle it will not hold the hinge screws very well if at all. Pull the trim off the existing doorway and see what it would take to install a pre-hung door. Allow 2 extra inches on the width for the new door. 32 inch door need 34 inch by 82 3/4 inch rough opening. 30 inch door needs 34 x 82 3/4 opening.