Can ceiling fans push heated air down?
Yes, ceiling fans can and do move air downwards, regardless of which direction the motor is spinning. The direction of the motor determines where the downdraft air will be coming from, as well as its relative temperature:



In the summer, you set your fan to spin counter-clockwise, so that air is directed downwards underneath the fan. This provides a cooling effect on your skin, and the air being forced downward is relatively cool.



In the winter, you reverse the fan direction. This directs the relatively warm air above the fan outward and then downward, making the rest of the room feel warmer. 

There are certainly some fans on which the motor speed can be reversed using a remote; just search your local fan store. However, this site indicates that fans mounted on cathedral or vaulted ceilings are too high to produce a wind-chill effect and should be left spinning CCW (in "summer" mode) year-round (although it is the CCW direction that produces the wind chill effect, so maybe a typo?). Another site indicates that high ceiling fans should be reversed. The US government's energy star site doesn't take a position either way, so you may just have to try it and see.

In any case, ceiling fans only cool people, so there's no need to run them when nobody is in the room. And their effect is felt pretty quickly after they're turned on, so no worries about having them on before you get into the room.

Fans installed over tables are probably the exception to the above rules, as you'd always want them running CW. This prevents the downdraft from being right over the table, cooling off food too fast or blowing homework, playing cards, etc around.



Images taken from here