If I have a weapon in each hand, but swing them one at a time instead of using that lame little auto-combo, does it make each individual attack faster and stronger if I have the right perks? For example, if I power attack with my right, quick attack with my left, and then quick with my right, do the perks still work?

http://uesp.net/wiki/Skyrim:Power_Attack#Dual-Wielding

You can do only one at a time I think. Its either left hand, right hand or combo.