I have the same issue. If the screen doesn't turn on and the lock screen or launcher doesn't rotate, I have to re-dock.
I've also had similar problems getting the HDMI detected. That one is extremely touchy. Sometimes once it's docked, I have to wiggle the phone to get the HDMI icon to appear so the WebTop will activate.
Does the hdmi doc only have an insert for the extended battery? I have the standard dock and the car dock. Both appear to be designed to work with an extended battery, so the stock Bionic doesn't fit tight. I have to make sure that the left side of the phone is flush with the dock as I am pressing down for it to work properly.
The HD Dock's fit is the same as the standard dock. The only difference in the interface between the phone and the dock is the HD dock has an HDMI plug on it. It does make the phone fit much tighter though. In my case, too much so. I feel like I am going to break something sometimes when I try to pull it out. I also have a standard dock and with it secured to my nightstand with Velcro, I can remove it with one hand easily. The fit still feels fine though.