I have my free hotmail account synced in my phone.
Firstly, your phone will "poll" your email account using SSL, to find out more in this just google SSL.
Secondly if you load your email and see the contents if your inbox, press the options button in your phone, (left hard button, not touchscreen) and select account settings from the menu. Here you can change how often your phone updates your inbox.
There also an option to set the account to delete emails from the server when you delete then via your phone. Although I'm not sure if this actually works on a free account.
If you change your password on a pc, your phone won't be able to poll your account it'll the phone is updated with the new password.
The email app will poll the account at an interval you specify, so if set for 15 minutes, at 3pm the app will poll your account, then it will be idle until 3:15 when it checks again, and so on.
The data downloaded when the email app polls is minimal, just an update to the counter to show you if there is anything new. If your in the inbox view then you get what you see, sender/subject info. When you open the email, that entire email is downloaded, I believe.