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Question 1: Given a number $N$, can we construct a convex planar region that can be cut into $N$ mutually congruent, connected, convex pieces but not into any other number of connected, mutually congruent convex pieces?
Partial Answer (guess): For prime $N$, there seems to be a simple way. Take a regular $N$-gon and mark from it $N$ mutually congruent quadrilaterals by drawing lines from center to mid points of the N faces. Now in each quadrilateral, replace the two 'outward' edges by copies of a polyline with say $p$ edges and with angles that are irrational fractions of $\pi$ (see ref 3 for some justification for 'irrational') in such a way that the $N$-gon becomes a convex $Np$-gon. This $Np$-gon seems to allow partition into $N$ and only $N$ pieces that are mutually congruent, convex and connected.
Remark: As per the answers below, one can upgrade above answer to work for all values of $N$, not only primes.
Question 2: Are there convex planar regions that allow partition into mutually congruent and connected pieces only when the number of pieces is one of exactly $2$ specified values — for example is there a convex region that can only be cut into $3$ connected congruent pieces or $5$ congruent pieces but not into any other number of congruent pieces?
Remark: Answer to question 1 can be slightly modified to yield planar regions that seem to allow partition into only $N$ mutually congruent pieces or $kN$ mutually congruent pieces where $N$ and $k$ are primes.