First thing, you should do, if you haven’t done so yet, is to generate fresh new Yii base application, using yiic (and templates) that ships with new 1.1.13 version. Then copy pieces of code that causes problems (jQuery code delegation and bootstrap menus) to see, if you can reproduce the same errors and problems in fresh application.
If yes, then you may assume, that problem does lies in Yii. But most of such cases shows, that freshly generated application doesn’t have these problems, which most likely means, that problem is inside your own code, not necessary there, where you’re looking for.
Test freshly generated application (as well as your own) on both localhost and some remote server. I’ve seen enough situations, where problems were caused just by a different server configuration, that I’m suggesting this every time ever since.
Silly question: You do purged assets folder of your application after upgrading Yii to 1.1.13, right?