Is every free monoid a pure monoid and vice versa? A pure monoid is a monoid where only the identity has an inverse. Is every free monoid pure, and conversely?

Yes, free monoids are pure. By definition a free monoid on a set is the monoid of words consisting of elements from that set, and there is no cancellation, so no inverses, apart from the empty word.

The non-negative rationals under addition is not a free monoid, but it is pure.