There's usually a solution, but it ultimately is pretty tweaky depending on the exact nature of the plug-ins you're using.
It sounds like your existing scripts are dependent on an earlier version of jQuery than that used natively by WordPress. If you're enqueueing an earlier version of jQuery, than it's also likely that those menus and sliders are expecting jQuery not to be in no-conflict mode - which is the WordPress default.
So, you'll have to either tweak your current scripts to support the standard WordPress usage of jQuery (current version, no-conflict mode), or you'll have to tweak My Calendar so that it's compatible with conflict mode and jQuery 1.4.2.
Either way, you've got to tweak something - and without specific knowledge of the issues, I can't really give you any specific advice.