Regarding the Shut Down freeze (mentioned here before as a Mac OS 7.6.1 problem), Sy Kallhof writes that, after installing Mac OS 7.6.1 on his PowerBase 200, he would get a Shut Down freeze every time he selected Shut Down from the Finder's Special menu. However, when he selected Shut Down from the Apple menu, no freeze occurred.
d. Apple Menu Options Specifics are toovaried for me to want to go into details here, but I have gotten several reports of the Apple Menu Options control panel as a source of some of the freezes and slow downs associated with Mac OS 7.6.1. Checking this out is easy enough to do:
Disable the control panel and see if the problem goes away. If so, delete its Preferences file before re-enabling it. If that doesn't work, reinstall a fresh copy of the control panel. If that too doesn't work, you can try fiddling with its loading order. Otherwise, give up on using it.