As far as drawing air into the house through the skylight, you are trying to reverse the natural flow. Heat rises, so trying to draw air in through that opening goes against the laws of nature and means you will have to work harder.
If your attic is vented well you can install the fan in the ceiling of the upstairs and push the air out through the attic. This will pull outside air into the house and push cooler air through the attic to reduce the temperature there as well, reducing the heat load against your insulation. I have owned a two different homes with whole house fans installed like that and they do a wonderful job, especially in the evening when outside temps are dropping. They obviously don't help the humidity levels. Add a set of louvers to the opening in the ceiling and seal the opening against cold before winter. I built a box around the fan opening in the attic and put an insulated cover on it in the winter.