Should door frames in the bathroom have been sealed with silicone?

The tile backsplash is approximately 150 high. The gap above corresponds to the tile thickness. Should this have been sealed?

The frame sits on the wall with me. Visually, something should go in there.

I would also have assumed that the frame belongs on the wall and the baseboard connects there. Or do you have a controlled residential ventilation system and this is an intentional gap for overflow?

The tiles do not belong under the frame.. botch.

No, no botching.
It will be done exactly as commissioned.

A good tiler, however, will point out both options beforehand.

And yes, silicone or acrylic should go in there or have the frame reworked by the carpenter.

Unfortunately, we had the same problem because the tiler came before the doors were installed and he couldn't know how far to tile. In our case, the frame was supported in the upper area with a narrow wooden strip. After wallpapering the upper surface, it was then filled with acrylic and painted over. Not perfect, but not really unattractive. Obviously, I would do it differently next time.