Comment by defrost

1. No. - https://en.wikipedia.org/wiki/Fixed-point_combinator

In a generic context a fixed point combinator will only return a fixed point of a function if one exists.

2. No. (see: 1. above)

Your confusion stems from the usual domain of application to function spaces where every function has a fixed point (ie. not all function spaces, just those that are made up of functions with at least one fixed point).

From the link above:

    In the classical untyped lambda calculus, every function has a fixed point.