Hovering Class Browser is a very good feature especially when one is working on someone else's files. I was however a bit disappointed when I realized that I have no other choice than using the mouse to display the details.
Similarly to the 'Show current scope when typing' option it would be great to have another kind of behaviour, say 'Show details also when typing' that would behave the same way as hovering but would follow the caret as well.
The caret and the mouse probably point to different positions, so the behaviour should be clarified well. I think the last touched one should be considered. (CTRL+down (and up) is tricky because it doesn't move the caret but modifies the mouse position relative to the text. I think this shouldn't be treated as a mouse movement, just like it works now.)
I don't think having it always enabled for keyboard would slow the editor. There is already a delay in it, after all the mouse can move over the text much faster than anyone can type or navigate.
In my opinion the possible candidates for shortcuts are:
HCB for mouse on/off
HCB for caret on/off
show HCB info for mouse
show HCB info for caret (your suggestion)
The latter two have two purposes:
control ambiguity (when you have moved the mouse, but are still interested in the stuff at the caret)
that mode is not enabled, but you are interested in it sometimes (your suggestion)
Your suggestions are reasonable and your reason as to why we don't have keyboard access is correct -- the "caret and the mouse probably point to different positions."