To answer question #1, you CAN use Outlook on one computer and Outlook Express on a second computer to check the email from one account as long as that account is not on an Exchange Server. The results won't be what you expect though. If you download the mail to the first computer, that mail will not be available to you on the second computer. Likewise, new mail downloaded to the second computer won't be available to the first.
To answer your second question, you can not just move/copy Outlook to another computer. It must be installed. If you do not have the installation CD, you're out of luck.
Take your new computer to your company's Technical Support and let them install Outlook for you. Of course, they're going to ask why you need 'their' copy of Outlook on your personal laptop and may not do it for you. Then again, they might. ;)