How to get a shorter rod
It hangs a bit low

It will be 1/2" pipe or 3/4" Pipe, may need threads on your fan, You can buy various sizes from Home Depot. Or
You can, determine new height you want, measure the difference, that is how much you need to cut off, you will need to drill for the pin and thread for ground. Do not cut with wires in pipe. Don't use Pipe Cutter, will leave burr and mayby cut wire.

Correct. Some fans can be flush mounted.

Frequently asked ceiling fan questions - Ceiling Fans N More