Electrical – DAC Output to 4-20mA

To answer one of your questions, this DAC (I've added a link to your question) has no internal register, so yes, it is constantly converting whatever 10-bit value that appears on its inputs to the corresponding analog value.

Usually, the current spec of a output pin is so that positive values denote current flowing out of the pin. This should be clearly defined in the datasheet.

In your case, it's obvious the pin is sinking current, meaning this "center current" is flowing into the pin. Given no other information, this is what you'd expect from the negative value, but it's also clear that is what is what is expected by the example circuit. The opamps are connected as inverting transconductance amplifiers, with current flowing into the D/A pin causing positive output. The voltage of these D/A pins will be held at 0 by the opamp circuits.