Yes: There are definite advantages to fall fertilization of hardy plants. But, as with all fertilization you should sample your soil and have it analyzed to see if you need to apply any fertilizer at all. Your state Cooperative Extension Service should be able to help you with this.

Yes you can do both fall and spring fertilizer on deciduous plants. It just depends on a lot of things but you will get a lot of growth out of both fall and spring. I think root growth is at its peak during late summer through fall (or winter if ground doesn't freeze). Usually, it's the nitrogen that is lacking. If it's sandy, you tend to lack certain nutrients. If it's clay, it tends to be rich in nutrients and the only thing you needed is nitrogen only. Like bboy says, get soil test assuming you're using synthetic fertilizer. Organic fertilizer is different requiring different type of soil testing like Texas Plant and soil lab in my case.