There's no functionality for that. People don't typically hover over a field label, but the certainly hover/click/focus on the input which is why strong UX is to show it there. Hover only is not responsive compatible (will not work on mobile devices) and taking over label click behavior likely will break focus on click behavior among likely other things.

With that said have added a feature ticket to see if adding an additional opener trigger for tooltips is possible so hovering label would trigger the input tooltip to display.

forge.joomlapolis.com/issues/6441

I guess there is also no way to tell the field description to show up permanently unterneath the field.

That's doable under the layout parameters of your field where you can override the icons display. You'll notice modes for "Info icon" and "Info text" the "Info text" modes output the description as just a string directly next to the input. You can use CSS to move it below the input. You could also use the profile edit value layout parameter to further customized the edit display of an input.