I know it can be tough to change email accounts but I had to do the same a number of years ago. My old POP email is forwaded to my iCloud account, although I gave everyone my new address and it wasn't long before no one sent email to my old account. The POP system is quite old and outdated now, and was desiged when people only checked their email from a single device. Most people have transitioned to IMAP accounts at this point.
You could forward your email to either your GMail account or your iCloud account. They are both IMAP accounts and will provide the behaviour you're looking for. However, when you reply to a forwarded email it will be show that it was sent from GMail or iCloud, not from Hotmail.
If you want to move your contacts, you can download the app My Contacts Backup and use it to back up your contacts as a vCard attachment to an email that you send to yourself. This attachment can then be used to import them to either GMail or iCloud. If you want to import them to iCloud I can tell you how to do that.