You can only use Hotmail with your iPhone "Mail" app, via POP3, which isn't a true sync service for email. All it does is allow a copy of mail to be sent to a remote mail client. As such, there isn't two way sync - so even tho you mark mail as read in Hotmail on your computer via a web browser, the iPhone mail client won't be told that. Likewise, when you send an email using your iPhone, that won't show up in your Sent folder when using a web browser.

TBH, you're best off ditching Hotmail - it's old tech now, and not very user friendly IMO. Get yourself a Gmail account and use that instead. It's SO much more advanced and user friendly, and makes email so much easier to manage. Plus, the iPhone works nearly flawlessly with Google services, and you can set up gmail on the iPhone via either IMAP (which works brilliantly) or via Exchange, which works almost aswel, but also allows push notification for gmail (which IMAP doesn't)