I would appreciate if TAdvOfficeHint had HideAfter as TBalloonHint.
TBalloonHint aligns the window to the center, which is not nice. And he doesn't use / respect styles.

And TBalloonHint bold font is not a good choice

TAdvOfficeHint is based on the VCL hint system (overrides it)

so the settings Application.HintPause, Application.HintHidePause apply to TAdvOfficeHint as well.

How can I make the hint appear while the cursor is on the component and disappear at the moment the cursor is outside the component?

As explained, TAdvOfficeHint is based on the Delphi built-in hint system, so display and hide of the hint is the same as a regular Delphi VCL Hint.