Clearcast is chance to cast spell for free and reduce cooldown by 100% (though it used to just be cast spell for free ).
Specialist is chance to cast spell when casting spell (almost like how a proc works but on your actual skill). If say you used meteor once, the specialist makes you throw another meteor then. I think the second cast is free though or cost some mana.
Mirrorcast is similar to specialist but for all spells instead of the spell you cast as description says. They both double cast.
Will double cast affect KnightsCharge and whirlwind? I'm not too sure but it probably can. Clearcast on whirlwind and KnightsCharge, well it isn't like a proc but you'd only notice by lack of mana usage .