What is the purpose of C3? Is it to provide a path for high-frequency signals?
Basically. You'll see that referred to as a "feed-forward" capacitor in some datasheets, and is often present in converters that use a simple resistor divider for feedback. It serves the same function as the capacitors that you often see in parallel with op-amp feedback resistors, essentially making the feedback more responsive to higher frequency changes in the output. When sized correctly, that stabilizes the system by suppressing higher frequencies that may otherwise result in oscillation.
What kind of bandwidth requirements would there be on the op.amp?
It depends on the converter frequency. The op-amp circuit is in the feedback loop of the converter, so any lag it introduces becomes a phase shift in the converter's control loop, which will decrease the stability of the converter. This can be somewhat compensated for by C3, which effectively slows the converter's response to increase that phase margin at the expense of how responsive the overall loop is. Note that the effective bandwidth of an op-amp circuit depends on its gain, so larger gains require higher bandwidth op-amps.
If you don't need to have one end of the heater connected to ground, you could theoretically remove the op-amp entirely. Instead, you'd swap the current sense resistor and the LED, and take the feedback from the point where the LED and sense resistor connect via a resistor. The DAC would connect to the feedback pin via a second resistor to adjust the current (See figure 4-1 in the appnote, and just replace Vref with the DAC output). The problem with that approach is that the voltage across the sense resistor needs to be around the feedback voltage on the converter which can result in significant power dissipation.