You should be able to. There are track adaptors which will take a basic cord for a pendant or a plug in. I've linked to the plug in version. Look for EP 814. If you go to the next page, the EP 815 can work as well. You'll just have to cut the actual plug off and wire your fixture directly into the 815. This one is nice since there's a ring to support the weight of the fixture. I don't know if your light has a support cable or not. You obviously don't want it hanging just from the plug with no support. Here is a link that might be useful: Plug in track adaptor

Sorry, but I've never seen in done in any fan or lighting catalogue. I would suggest that it shouldn't be done

if it happens shortly after the fixture is turned off after being on for a time it may be something cooling down (incandescent bulbs produce some heat). I seriously doubt its an electrical problem.

It seems like it should be possible to use the glass shade from your light fixture on a new ceiling fan, provided you choose one with the right size fitter. Here's an article that addresses changing the glass shades on a ceiling fan: http://homeguides.sfgate.com/upgrade-ceiling-fans-21435.html