Fix mobile overlay spacing and home action placement

[Teeldinho]

Summary:

• Separate the mobile run/jump overlay from the contextual action panel so prompt width changes do not move gameplay controls.
• Place the home action to the right of HP on tablet/mobile and at the far right of the desktop header beside settings.
• Center the mobile home hero status copy on smaller screens.

Validation:

• npm run lint:fix
• npm run typecheck
• npm run test
• npm run ci:local