If your trunks are PRI then 'one number (main bill) vs DID' is under the controll of the PBX (generally). If you have 'plain' T1's or analog trunks, then CallerID is under the control over the carrier, either local for 'local' trunks, or LD for LD trunks (assuming you have separate).
If you have PRI trunks, and if your CLID tables are set up correctly as John has suggested, and if your carrier allows you to transmit something other than your main number (configurable at the CO), then yes, you can transmit a specific DID number for every call.
The name is not stored in the PBX, the name is in telco records of whatever telco you get your DID's from. In the US, name is 'retrieved' from the far end by the _receiving_ carrier using a database lookup. Typically the 'name' on a business account is the same for all lines on the account, and is the business name (xyz college).
Verizon and SBC aren't playing nice in Kansas City, so for example, if I call from my office in Pennsylvania to my uncle in KC (verizon residential to SBC residential), he gets my number because the number is sent 'forward' with the call, but he gets 'pennsylvania call' because SBC won't pay Verizon to lookup my name.
I honestly think what you need to do is call your LD carrier and tell them you want a static calling party number of xxx-xxx-xxxx on your long distance T1, and that number is your main switchboard number. They shouldn't argue.
I used to manage telephone services for multiple colleges and universities in a former job, so if you need help, just post here for me to call you: don't post your phone number or email. I think I know where you are.
If someone really wants to go to the 'extension' level for transmitting CallerID, then you have to have PRI trunks for all routes, have the CLID tables built correctly (don't forget to have excpetions for the president, dept heads, etc) and then work with the carriers to make sure their screening tables allow you to send your own CallerID. In some cases PRI is cheaper than other trunk types, but that varies from carrier to carrier and market by market, unfortuantely.