How to measure current without shorting/how to limit current?
DAQs are pretty standard
5V input logic low is also pretty standard &lt;0.8V for HCT 
Using active low,open collector out from DAQ, connect 1-10K pullup to 5V and connect to 5V logic input. This is 1mA.


Other details, not likely needed.


The easiest way is assume fixed output resistance and measure voltage drop or rise from the rails using 1K load to the opposite rail for 4mA range. Then compute offset V/1K to determine internal R.
Often current load for logic depends on TTL thresholds of 0.8V and 2V which limits current of 4mA due to 200 Ohm internal resistance of driver.
If wanting to indicate on an LED, then 3mA Is feasible with RED, 



  using Rs=(5V-2V-0.8V)/3mA = 730 Ohms or more.



subtract ESR of driver and 15R for 5mm LED and Vf of LED from Vcc to compute RS.