[Sylvain]

You iron is sharp (shaving your arm) but, ... while stropping, you might have changed the angle at the very tip of the blade with the following result: no relief angle.
If there is no relief angle, the blade will skate on the wood without biting in it (unless, maybe, if one pushes very hard vertically on the plane, flexing it).
P.S.: "The bevel must be [added: at least] about 2-degrees under the bedded angle of the iron for what’s called relief, that’s all. Otherwise, spring back in the wood’s surface causes the plane to ride the wood and rise up against the cut, resulting in a no-cut plane. "
In this post :
"In reality, however, we discover that a bevel on the underside of the cutting iron that is level and coplanar with the sole will not cut well or even cut at all. Why? For two reasons. One, for the cutter to cut, there must be a relief to the underside and two, edge fracture occurs in minute degrees right along the cutting edge as soon as the iron cuts the wood. The more you plane, and the more knots you hit, the more the plane iron dulls. The angled relief allows the cutting edge to engage in the wood. Take any plane type, the router plane, the side rebate plane, bevel-up planes of every type and even spokeshaves, there will always be a relief angle. "

Looking at Paul Sellers sharpening an iron, he is doing a convex bevel but while stropping, he is very attentive to not change the angle of the iron on the strop.

 

[Sylvain]

I'd start with sharpening to a higher grit. 1200 is generally a staring point.

Sharpening at 250 grit

 

[Sylvain]

Planing edge on some pieces was not a problem, [...]. If I try planning the wider surface (true face) the plane just bounces off the surface, as the wood was glass. Taking some cuts here and there, but no way of taking a shave longer than 30-60 mm.
[...]
If I put the blade deeper, it will stuck in the wood. Too away from wood - doesn't cut at all.

This is really typical of no relief angle.