Must a curve of constant width be generated with an odd number of sides?

As far as I understand the questions, you mean by the "Reuleaux-method" that you take a polygon and add circular arcs to obtain a convex body of constant width. Assuming this, the answer to question 1 should be negative:

In the book "How round is your circle" there is another method to construct convex bodies of constant width: See on the How round is your circle webpage, the example titled "Half a convex shape". In this example, half an ellipse is completed to a convex body. So for curvature reasons this body can not be generated by the "Reuleaux-method".

Just for fun, I made a little animation of the half-convex example linked to by @Alexander Schmeding. The upper half is an ellipse with minor-to-major axis ratio ranging from $b = 1/2$ to $b = \sqrt{2}$. The curve can be parametrized as $$f_b(\theta) = \begin{cases} (\cos \theta, b \sin \theta), & 0 \le \theta \le \pi, \\ {\displaystyle \frac{2 (b \cos \theta, \sin \theta)}{\sqrt{b^2 \cos^2 \theta + \sin^2 \theta}}} - (\cos \theta, b \sin \theta), & \pi < \theta \le 2\pi. \end{cases}$$ In Mathematica, we can generate an interactive plot with

F[t_, b_] := Piecewise[{{{Cos[t], b Sin[t]}, 0 <= t < Pi},
             {2 {b Cos[t], Sin[t]}/Sqrt[(b Cos[t])^2 + Sin[t]^2]
               - {Cos[t], b Sin[t]}, Pi <= t <= 2 Pi}}]

Manipulate[Show[ParametricPlot[F[t, b], {t, 0, 2 Pi}, 
                PlotRange -> {{-1, 1}, {-1.5, 1.5}}, PlotStyle -> Black], 
           Graphics[Flatten[{Opacity[0.5], {Hue[#/Pi], 
           Line[{F[#, b], F[# + Pi, b]}]} & /@ (Range[n] Pi/n)}]]]
       {{b, 1}, 1/2, Sqrt[2]}, {n, 1, 75, 1}]

And we get this:

What I found really interesting is how the curve looks like it's the same near the extremes of the animation, but it obviously isn't so from the definition of $f$ itself. Bonus points if you can parametrize the envelope of normals for $b \in [1/2, \sqrt{2}]$. And more bonus points if you can compute the enclosed area.

It depends on what you mean by the number of sides of a curve.

If $k$ is an odd integer, and $p(t) = a\cos^2(kt/2) + b$ then $$ \begin{cases} x(t) = p(t)\cos t - p'(t)\sin t \\ y(t) = p(t)\sin t + p'(t)\cos t \end{cases} $$ where $0 \le t \le 2\pi$ is a curve of constant width. Choosing $k=3$, $a=3$, $b=1$ you end up with

How many sides does that curve have? More details can be found in this document