Does eM Client work like Windows Mail in that messages remain on Windows Mail server?
EM Client would work with any mail server that supports IMAP or POP3. By default IMAP e-mail are left on the server until you delete them (from both web and/or eM). POP3 server however can set the setting in eM not to delete e-mail. But some POP mail server has a validation period, where the mail expunge itself after marked retrieved by POP protocol. This is entired server side.
I haven’t use Hotmail/Live for over 12 years. I know Hotmail/Live only support POP3 and Microsoft’s own protocol. So I’m not sure whether mail expunge itself, you need to check this with them, as this is out of eM Client hands.