Does this modem support DTR dialing?

Yes, this modem supports DTR Dialing $D(command).
The DTR dialing method is popular in synchronous applications. (DTR) is an input signal to the modem from the terminal or computer to which it is connected, on pin 20 of the RS-232C interface. In DTR dialing, the modem automatically dials a stored number in the N0 location of memory as soon as it receives a high DTR Signal.
$D0 disables DTR dialing.
$D1 enables DTR dialing
DTR must remain high for the duration of the call.
Make sure DIP-Switch #1 and #8 is in the UP position (12 position switch).
Store the phone number in the N0 location with the command ATDTxxxxxxxxN0