Trisecting
a Möbius strip results in two strips. The smaller has one half
twist and is identical to the original strip. The larger strip
is twice as long and has 4 half twists.

Bisecting
the smaller band would give just one band. The larger band (with
the 4 half twists) will become two bands when bisected.

Yes.

Solution

No.
One of the two resulting pieces is equivalent to the original band.
The other is a different structure.