Is it possible for a motion to be isochronous (time period is independent of amplitude) but not true s.h.m.? Can an s.h.m. be non-isochronous?

Well, assuming that by s.h.m. you means "simple harmonic motion" then to answer your second question first, of course all s.h.m. systems are isochronous.

Your first question has also a simple answer: a point mass submitted to a uniform gravity within a cycloid shape, also known as "tautochrone curve" or "isochrone curve" has a period independent from the amplitude, without being a s.h.m. https://en.wikipedia.org/wiki/Tautochrone_curve

Finally to your third point : no, you can have periodic motion that do not have an s.h.m. approximation at small amplitude.

Consider the curve $y=x^4$, $y$ being the vertical direction and $x$ the horizontal one. Put a ball in it, without friction. The motion will be periodic, but the period of small oscillations will increase as the inverse of the amplitude in the $x$ direction when this amplitude becomes small (as a consequence of the virial theorem : kinetic energy that behaves as (X/T)^2 of the same order as the potential energy that behaves as X^4 hence T behaves as 1/X, where X is a typical value of $x$).

(This is true only at small $x$, when the motion along $y$ is very small. For large displacement, the period goes like $sqrt Ypropto X^2$, $Y$ being a typical value of $y$, but this is a completely opposite situation.)