I hope an example schematic will help you to understand better the use of the Open collector output.
1. Pin4 is used to connect external voltage to drive the connected relay.
2. Pins 5 (COM) and 10 (NO) are rather output pins of an internal relay to power externally connected devices. A couple of examples below. 5 and 10 contacts also have different electrical ratings.
3. The parameter you should be looking for on the power supply is voltage. If voltage is sufficient, the circuitry will draw the amount of current it needs, as long as the supply is rated to provide at least that amount, so you should be able to use the power supply the device shipped with.
4. Due to internal circuitry your proposed wiring will not work.