E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.
Poincaré Conjecture is yeah interesting to explain though to be honest I vaguely recall an article in similar style to your post that explained it.
Possibly something like describing gradient descent on a manifold is interesting to this audience? Or maybe a post on Flatland? Many possibilities really on good follow ups.
Can a point be a manifold?
(not that there are different 'classes' of manifolds out there; I am not sure if the point would qualify as a Riemannian manifold, for example)
Some are. A Koch curve for instance.
It depends on which mathematicians! Plenty of differential geometers allow manifolds to have boundary, and say "closed manifold" (https://en.wikipedia.org/wiki/Closed_manifold) to emphasise when they are dealing with a (compact) manifold without boundary (or, as you point out, really a manifold whose boundary is empty).
f(0) = 1