The last time I drew mouldings I had only to trace them with a polyline and extrude. Now even in 3D hidden I can see exta lines, as if this is merely the shell of a molding and not a solid piece. How do I make a solid from a profile?

You could "list" the object and find out if it really is a "3dsolid". if it is, then it's just a display setting.
 
Hows it look in another shademode, such as "realistic" ?

First you need to make sure your profile is closed. Type 'pedit', select single or multiple. Enter an option: 'close'. Enter, Enter, done: you now hav a closed loop to extrude.