I'm not really referring to the amount of skills you can equip, but mainly that there should be restrictions to WHAT skills you can equip. The current max in skill slots does need to be increased, maybe just simply, another row on the action bar.

Thats pretty simple enough as it is. When talking about cross-classing, as you said, "doubleing the cost will impossible to use your own classes skills", is the whole idea behind what I was talking about in the first place...
There needs to be a cost as to what you give away out of your own class, in return to have something from another class.

If you want to make a "pure class" then just simply never use other skills from other classes.
The element of doubling action costs for equipping skills is just a variable, obviously there doesn't need to be such a restriction for it too work either way. Simply you could just say, "X" skill is transferable, "Y" skill is class specific.

Balancing out these many factors is the fun part of growing, and customizing something you want to play.