I agree with Paul.
Make a full pot of coffee, get some scratch paper, sharpen about 10 pencils, get out all your reference books for IE and Outlook, sit back and poke through the preferences / settings for a couple hours / days / weeks
until you stumble acros the right tweek to disable all the off-line features you wish, plus the "Turn Off Modem If Idle For More Than XXX Minutes" settings (typically found within the Modem's Properties).
Not being a heavy IE user, I cannot suggest any more than the above - or at least hunting around the "Internet Settings" found in the Windows Control Panel.
Used Outlook at work, and it was a major quest trying to locate where to apply each tweek. Hunt around the File Menus (top File menu bar) for awhile until you locate what's needed.
If you are unable to fix anything via this "Find It By Hunting" technique, I can dust off some books and see whaddaheck to do.
Let us know.
P.S. Any hard-core IE / Outlook users out there?
Toss in some helpful hints!