IF its a dead screen, you can plug in an external monitor and hit the monitor keys in the blind until you have pix on it.
No, bookmarks don't take up cache (RAM) memory, but having them all open does. Firefox allows the creation of nested bookmarks in a folder that you can then open all at once under the same icon in the toolbar. Between my RV, RC, FCC bookmarks all showing up as 3 icons on the toolbar, I have over 30 individual pages open in the background. THAT is what runs you out of RAM. The IT guys call me 'special' and don't answer the phone when I call anymore.... :(
Buy as much RAM as your new PC can stand to run and its likely you will never see that flag again.
For exporting bookmarks, look on the Bookmarks or favorites for Export. You should be able to save the entire file and restore on the new machine exactly as it looks now. Im not an Explorer fan at all, so don't have allot of experience in how to move them. But FireFox's bookmark can be imported into and read by a Mac as well.
