I thing you understand what server means. I assume you have a POP account. Anytime you add a POP account to a different computer, it will download every message from the server. There is no way around that unless you call the provider and ask them if there is something they can do. Storing messages on a server in a POP account is not a good idea in my opinion. Others may differ on this. Unless you have a paid account, there is probably a limit on how much you can store. Maybe hotmail doesn't have a limit. But you will still have to download all the messages anytime you connect the account to a different computer.
Downloaded messages are stored in a file as they come into the account so they don't get redownloaded each time. Since that file doesn't exist on a different computer, it will treat each as a new message that hasn't been downloaded.
So, if you want to leave them on the server, let them download. Use the smart mailbox to locate the older messages and delete them. A better option is to create a local mailbox on the computer and move them to it. If you have a backup plan, then they will be backed up. That way you have an additional copy.
As for the other question, I don't understand why it didn't give you the option to create the email. Are you sure you don't have one? Try signing into iCloud.com website and open Mail there. See if it gives you the option to create an account. If you already have a me.com address, mail will open there and you can see what it is.