I have two questions. Does wearing Heavy Armor affect move rate, attack speed, or dexterity negatively? If I have some Heavy Armor and some Light Armor in my set, can I use skills from both trees?
Of course you are right with the innates but this is just the case if you are training these skills. I think a blank avatar will see no difference regarding move rate, attack speed or dexterity if he is switching between heavy and light armor.
Heavy armor mainly affects your fizzle chance. No. You can only use the tree of the chest piece that you're wearing.
1) It does not affect it "negatively". Light armor has passive skills that increase your attack speed and move speed, however. Along those same lines, you can masterwork dexterity onto light armor while only being able to masterwork strength onto heavy armor. Enchanting is unaffected. So while heavy armor doesn't inhibit you, light armor has provisions to passively increase those three attributes with training/gearing. 2) As stated above, the chest piece is all that matters for considering if you're able to use skills/passives from the heavy armor or light armor skill trees.