I am currently replacing an old fluorescent light fixture with three
recessed lights.

A 20 amp fuse (not breaker) was used for the circuit that contained
this fluorescent light and the over-the-range microwave. The
previous-to-previous owners did much of the work in the house
themselves and usually did a poor job at it (for example, after taking
down some drywall we discovered rot everywhere that lead to finding out
the the overflow pipe that connects to the bath tub was never fitted
with the rubber gasket so there was an inch gap that lead to the tub
leaking whenever a bath was taken - they instead CAULKED the overflow
outlet in the tub).

In this situation the main feed from the fuse goes somewhere (sorry I
can't be more helpful); an outlet box that is about 1/2' from the
ceiling (weird I know) has a 3-wire cord (red, black, and white)
coming from the ceiling and contained a splice to a 2-wire cord (black
and white) that connected to the fluorescent light (with electrical
tape). I ripped all the tape apart and removed the outlet box as we'll
be permanently removing and filling in the hole where that was -
unfortunately I didn't look to closely how the wires were connected.
From this they must have had two of the three from the 3-wire cord
spliced together with one from the 2-wire cord. I removed the 2-wire
cord and all that is left is the 3-wire cord that I hope to use for the
three recessed lights.

I took a multimeter to the three wires and this is the result:

SWITCH OFF
black & red = 26V
black & white = 120V
red & white = 18V

SWITCH ON
black & red = 0V
black & white = 120V
red & white = 120V

A single switch controls the light fixture but the microwave is always
on. Unfortunately I cannot see behind walls to well and I am not sure
how everything is connected, but if persuaded I'll delve deeper.

Any ideas on what is going on? Is the black and white used together
for the microwave but is for some reason still fed up to the light
fixture? What do I splice together out of the three to form one hot
and one neutral?