How do HDDs maintain flying height (head-platter distance)?

Solution 1:

But in that case, what is the position of the slider when the drive is not spinning (head is parked)?

The typical modern HDD will always (try to) reposition the R/W head assembly on shutdown or power loss to a safe area of the platter(s) called the landing zone.
This is typically located between the innermost cylinder and the spindle/hub, and is an area not used for data storage. This scheme is known as Contact Start/Stop.

Another scheme used in HDDs is load/unload ramps at the outer edge of the platters that can keep the R/W heads from touching the surfaces.

Disk drives with removable platters have head assemblies that can retract/lift the R/W heads away from the platter surfaces, and move heads and arms away from the platters and off to the side.

And how it transitions from parked to "flying" above platter without contacting that platter?

On initial spin-up, the R/W heads are either (a) in contact with the platter surfaces, but in the designated landing zone, or (b) parked on the ramps. Only after the drive verifies that the spindle-motor/platters have achieved the required rotational speed is the R/W head assembly commanded to perform a recalibrate seek to cylinder zero.

As long as the platters maintain some minimal rotational speed, the air bearing should be sufficient (under normal conditions, i.e. no shock or vibrations) to avoid a head landing/crash on the platter surface.

But of cource the actuator is not spring loaded ...

Your assumption is incorrect.
The R/W heads typically have torsion or leaf springs on their arms so that the "normal" (i.e. parked) position is light pressure on the platter surface.
The spring tension can be adjusted to achieve the proper flying height of the R/W heads.

So where is the bug in my presumptions?

Seems like you have more than just one misconception.

Is there really a 5nm gap that is maintained in all drive states?

No, the flying height can only be maintained while the platters are spinning.

Is the air bearing not only pushing slider away?

(The negation in the question is superfluous. Don't try to assert your assumption while asking a question.)
The air bearing gives each R/W head lift (away from the platter surface) to counteract the opposing spring which pushes the heads towards the platter surface.

Related to this, when watching people unstucking the heads, they move the head across stationary platter (sometimes rotating the platter a bit simultaneously). Are the sliders in contact with the platter in that case or not?

The R/W heads would be in contact with the platter surfaces, but such minor contact should be inconsequential. The larger risk is contamination.

Addendum

A WD white paper on Contact Start/Stop versus load/unload ramps (PDF, 957KiB): Ramp Load/Unload Technology in Hard Disk Drives.