A few ohms for R710 seems about right. The gate drive is a push-pull:
Even though the datasheet shows a relatively slow turn on and turn off time, there could still be a bit of gate oscillation without a resistor here. I would suggest (as you note) something in the order of 2.7 to 10 ohms as a starter; there is indeed a trade-off between gate slew and gate ringing.
R712 is a series resistor into the current sense input (current limiting is set to engage at 260mA according to the datasheet). I think R712 is there to provide a helper filter so the leading edge blanking can operate properly; it is not that uncommon for leading edge blanking to get 'confused', depending on application specifics. I would assume that the first pass of the design had some anomalies around this area (there is an internal chopper circuit).
It is difficult to assess the specifics of this resistor, but something around 33 ohms might be a good starting point, although I have not done a full analysis, so treat this recommendation with caution; it is where I would start for a leading edge blank filter.
I agree with 'R703' probably being a 24V device (the controller is rated at 36V).
Excellent job with the schematic.
[Update]
The part has a fixed leading edge blanking time, derived from the internal oscillator apparently, as the resistor used to set it is a parameter on the datasheet line:
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A fixed blanking time could conceivably be an issue depending on the specifics of the design , so it quite natural to see a resistor here that can form a small filter (because the blanking is too short in a given design) in conjunction with track and pin capacitance (and possibly internals that we have no knowledge of).
From that perspective, it is quite feasible that the filter resistor is several hundred ohms to even a few k ohms, as noted by Nick.
Best Answer
Common problems with these power supplies I've encountered are:
The resistors in question are the 'kick start' - they give enough voltage to the switcher ic to start up. Once started, a winding on the main transformer is used along with a diode and capacitor to supply the switching ic from then on. The capacitor (usually 47-100uF 25-50V) can dry out. The power supply normally hiccups in this case.
Output diodes gone short circuit. This will shut the supply down quick sticks. Measure the diodes. Usually these are TO220 pkg dual diodes for the main high current output.
Main input capacitors drying out. If this happens with a psu topology like yours, then the switching transistors vapourise. You can replace the transistors but they keep on blowing until you replace the electrolytic caps.
Note that some power supplies don't like starting into no load. Some big resistors or some automotive light bulbs can suffice for a dummy load.
Sometimes you get a random failure. In this instance it a matter of testing each component. You can test the switcher ic by powering from a bench supply - read the chip's datasheet to see what voltage it requires - usually 12-15V. Use an oscilloscope to see if the ic is outputting pwm to the transistors. Make sure the power supply itself is NOT powered by mains voltage!