Which is better between a built-in shower mixer or an external one?
It depends: built-in for modern design, external for convenience.
Is the built-in mixer difficult to install?
Yes, it requires masonry work.
Is the external mixer less attractive?
Not necessarily, there are design models.
Which one is cheaper?
The external shower mixer.
Can it be switched from surface-mounted to recessed?
Yes, but it requires work on the system.