I'm not sure about the favourite links problem, but I can help you with the first thing.

Go into Control Panel > Internet Options (or Tools > Internet Options from browser)

Click on the tab marked "Content".

You'll see a section marked "Personal Information" and a button saying "Auto Complete...". Click this.

It will say "Auto Complete for..." so uncheck things you don't want it to remember, for example if you uncheck web addresses it won't remember web addresses, and so on. For the search engine thing you probably need to uncheck form. That's it