There are a lot of good detailed answers here. Here is the simple clear cut answer.
Does an X client necessarily need a window manager to work?
No: But without if you will struggle to manage your windows (moving, re-sizing, lowering, raising, etc.). There are other tool that can do this, if you need it.
Can an X client work with only the X server?
If an X client doesn't have a window, does whether it can work need a window manager?
If it has no windows, then there should be no down-side of having no window-manager.
If an X client can work without a window manager, does the X client necessarily have no window?
No: it can have windows.
vncserver -localhost -geometry 1920x1080 -SecurityTypes None $DISPLAY &
then in the new
You may need to install