How can I speed up rate at which dock appears on second screen?
Try something like this in your terminal. 

 defaults write com.apple.dock autohide-time-modifier -float 0.3; killall Dock;


Where the number 0.3 is, you can insert the number of seconds you want the animation to take. I found 0.3 to be quick enough, but still provide a nice effect. 

P.S. grabbing this from Col1107's answer. Just wanted it saved.