1) That can be set up as a control map for your combatant.
Set up a control map key with the None action, which will end the turn.
2) Generally, you can add control HUD elements to any other HUD and even UI boxes.
The scene hierarchy context menu has ready to use setups that you can create: Makinom > HUD > Control
There are controsl for buttons, axis (single axis) and joystick (2 axes/input keys).
3) Might be due to the order of the components.
Try moving the HUD Click component above the button control component.
4) You can just use regular Unity UI components, i.e. just add an Image component and select the sprite you want to use.
The HUD content components are used to display specific content coming from the HUD's user/content, e.g. a combatant's name or icon, equipment content, etc.
So, as long as you don't need content-based display that can change due to what's displayed by the HUD, you can simply add a sprite or texture using regular Unity UI components.
5) The button control doesn't really use a Unity Button component. Unity UI states are handled differently, so it being selected or highlighted can change due to other UI (e.g. a UI box having some button selected).
Try the setup that you can create via the context menu.
6) Is this with HUD click on the same game object as a button HUD? If yes, try as explained in #3.
Otherwise, might be the phase input key causing input reset before the click is used by the HUD.
7) As said above, try a setup created via context menu.
Also, is the input key working when using the actual key and not the HUD?