How much drainage pipe should I bury for a new downspout?
The length and size of your drain depend on several factors:


How many inches of rain do you get
What is the area of roof that will be draining
What type of soil do you have (sand, clay)?


For items 1 and 2, this calculator will tell you how much volume you have to be able to handle: http://www.calctool.org/CALC/other/default/rainfall

Sandy soil will dissipate water faster than clay soil.  Not sure how to measure this though.  You should probably allow a little extra if you have clay soil.

Typical slope for a gutter is 1/4 inch for every 10 feet.  I think you could do the same for your drain.

Yes, put gravel in the trench.  It's a good idea to wrap your gravel in a porous landscaping cloth that will allow water to pass through, but keep most dirt out.