If an Exertion result is rolled on an Initiative check, does it add a fatigue (or stress, as appropriate), or is it ignored on Initiative checks?

If it is applied and it was a Monster's initiative roll, then would the Exertion result mean a Wound for that Monster?

Yes and yes as far as what I would say

I will say yes and yes as well, just as I will rule that the universal effects of two boons or two banes apply to initiative checks.