Why the whine with straight cut gears?

Because of the impact of the straight cut teeth. With straight cut gears each tooth immediately makes full contact across its full width. With a helical gear the teeth engage and disengage progressively.

It's also worth noting that the only reason you use straight cut gear teeth is for strength, so once you've made that jump, most straight cut tooth profiles will be optimised for root bending (fatigue) strength, to get the highest strength. These profiles tend to have a higher mating velocity so are usually noiser. (i.e. it is possible to have fairly quiet straight teeth, but not at the same time as strength.....) So you could have quiet(er) straight cut teeth but it would be a bit pointless ;-)

I always thought that helical gears were stronger than straight cut gears. I thought that because the teeth were diagonal on the gear meant that they were longer, and therefore have a greater surface area, meaning the force transferred through the gears is spread out more. That's assuming the gears are the same size, obviously. Or have I just completely made that up?

As mentioned above, I also thought the main reason for using straight cut gears was to reduce the friction between the teeth, meaning less power losses.

Also no side thrust on straight cut. Helical cut gears will try and push apart, needing a thrust surface to hold them together...

You do still get separation force but it is radial not axial.