Does the load (or lack there of) have any effect on the temperature of the windings of the stator?
Unless the stator windings are made of superconducting material - the answer is very much Yes.
Temperature rise will be dependent on the current drawn by the load and the resistance of the stator windings. No load means no current, so there should be no power lost and, therefore, no temperature rise in the stator. As load current increases, so do the stator losses and the stator temperature will rise.
Is the output of the stator a function of its design?
Absolutely. However, like a great many electrical and electronic devices, heat is the limiting factor. One question is: How much headroom has the manufacturer allowed in giving the specifications for the design? Some manufacturers will cut it close and skimp on such margins to save a few pennies. The end result is that temperatures will rise to uncomfortable levels when high loads are presented - even when these loads are within the "specifications". Better made units will have heavier gauge wire with lower loss and, hence, lower temperature rise under the same load conditions.
That is to say the out put is just a function of design and RPM, so that you cannot get more than the rated output, so long as you stay within rated RPM.
Well, not really. You certainly can get more than the rated output - but the amount of the extra you can get and the time over which you can do this without destroying the unit will depend on a couple of factors - the main one being heat.
That is to say, one cannot burn out a stator by putting a load on it?
Yes you can. By putting an excessive load on the unit, you can certainly burn out the windings. On some of the cheaper built units, you can do this even when operating within the specifications. They just design it well enough to (hopefully) survive the warranty period.