Welcome to TRS
1993 to 2011 Ranger regular cab or extended cab beds will fit on your 2001, they are same 6ft beds
Exception was the Long Bed Rangers, few and far between, but they have 7ft beds and NO it will NOT fit on your frame, lol
In 2000 Ford change the tail light wiring in the bed to "2-wire" setup, previous years used 3-wire, so best bet is a 2000-2011 bed for the wiring part
If you find a 1993-1999 bed that works, you just need to swap over the bed tail light wires and tail light lenses they fit just fine
So not a big deal just a Heads Up
There are also step side beds in these years, they will fit as well, need narrower rear bumper, and tailgate, also shorter filler hose
Yes, the Mazda B-series beds in these years also fit, these all have fender flares on all 4 wheels, so have a good look at their sides to see if it will look good with your Ranger front fenders