Yes, you can send up to 2,000 messages per day. Outgoing messages must also conform to the limits of 10,000 total recipients per day (including the To, CC, and BCC fields) and 100 recipients per message. A large number of recipients can best be managed by creating a Google Group (mailing list) that contains those recipients and sending to its single address. For more information on limits, see: https://support.google.com/a/answer/166852?hl=en.

Sometimes you want to change the "From" address when sending email, but the address is not available. You can add an address that you own using the Google Web App. Log in and click on the gear icon in the upper right. Select Settings and then the Accounts tab. In the "Send mail as" section, first look to see if the address is already there. If it is, simply click on make default to the right of it and it will become your default "From" address.

Use a different browser to log in. Every internet browser (Explorer, Chrome, Firefox) stores login information — called cookies — that automatically directs the browser to the previously-used login page. Since the login page for a functional account is different from the one used for personal accounts, it’s often best to use different browsers to log in to a functional account and your own account.

While this is not a common occurrence, there are a couple of circumstances where this might happen.

It's possible that you've configured a message "filter" that is moving the mail. Filters can be found in several different locations, depending on if you are using a desktop client or the web interface. If you might have created a filter, simply remove or edit it.