No, the original password isn't stored anywhere for security reasons. But on the login page you can request an email with a special login URL that is valid for a limited time. After using that URL to login, you can change your password.
This forum has several features that can send you email, e.g. reply notifications and board subscriptions. The forum requires you to specify a valid email address and only send the required password to that address to verify its validity. This is necessary since otherwise a lot of people would type in bogus or broken addresses and try to use the email features anyway, which would result in hundreds of bounced email/error messages for the administrators. This registration process also serves as a double opt-in scheme, which prevents the forum from being abused for email-spamming other people.
You only need to logout if you are using a computer that is also used by other non-trusted persons. mwForum stores your user ID and password via cookies on your computer, and these are removed on logout. You can also use the "Limit login duration to browser session duration" feature in your user options to automatically logout when your browser's session ends.
If attachments are enabled in the forum and the specific board you want to post in, first submit your post without the attachment, after that you can click the post's Attach button to go to the upload page. Posting and uploading is separated this way because uploads can fail for various reasons, and you probably don't want to lose your post text when that happens.