Re: Email sync
Hotmail (old name for Outlook.com) is an email service. It is not 'web based', but rather can be accessed via web, pop3, imap4 or MS's Exchange system. The most usual method through MS products is via the Exchange system. The emails are stored on the MS server and you can download them to your PC through many email programs. If you set your email program to delete from both the server and the PC it will do both. If set just to delete from your PC it will do only as you ask. You can set the options on the account online (the server) to ignore your PC's requests to delete from the server too if you wish.
Assuming you are using an Exchange based client (Outlook or Windows Mail clients will usually default to Exchange) and you have not messed with the server settings it will sync, but not always immediately. If you sync manually you can force the issue. If you tend to delete emails then shut down straight away your PC may not have chance to sync. Also, when starting your phone, it will take a short while before it syncs in turn. Again, you can force the matter if you wish.
I suspect you have a very basic understanding of how email works OP, from your posts. You may really benefit from reading up on this topic in many ways. It is pretty interesting and useful. Best wishes on your voyage of discovery!