Some time has passed, and I've found some time to work in that stuff again. I haven't covered all of the points above, but at least some of them:

• The redraw doesn't happen so often, and if it happens, it doesn't kill the imenu-add-menubar-index.
• Speedbar now works for perl-doc mode.
• Searching for a module name in perl-doc now offers completion.

Version 0.81 will hit GNU ELPA within one or two days (wait for that, 0.8 has some nasty bugs).

I have prepared a short article about the news (using some markup not available on PerlMonks).