A minimum of SP4 is needed for BES to support Exchange 2003 SP2, there are issues with Calendaring in SP3 and lower for BES 4.0. If the BES is running on anything less than SP4 then you should apply SP7.
The BESAdmin does not need full permissions on the mailbox. If you delegate Exchange View Only Administrator on the First Administrative Group the permissions should inherit down then you will set Administer Information Store, Recieve As and Send As at the Server level. Then check that it is inheriting at the mailbox store level.
Send As in Active Directory is different, it's for an email issue. Send As in AD is needed after certain Exchange updates revoke the right of users to send email.
Send As should be the last one in the list after you have selected user objects. If you're having trouble finding it, use this article:
BlackBerry Search Results
Under Resolution 2 at the end there is a Tip with a link to a video on how to set the Send As permission in AD. That should be of help to you.