How exactly is IBU measured? Are IBU-testing kits generally available or do you require labs and lots of equipment? Is anything available for testing my homebrew?  (I am thinking of something like litmus paper)
Unfortunately there's no easy way to test for this at home unless you're a chemist and happen to own a gas spectrometer.  However, white labs sells a testing kit for &lt; $40 that will run a full series of labs on a sample of your beer and give you exact alcohol, IBU, diacetyl, gluten content, etc.  But it requires packing up a sample of your brew and shipping it to white labs in the provided box.  I don't know how long it takes since I've never done it.

As to the process I was able to find this description at homebrewtalk.com:


  A 10 mL pipet is dipped into octanol and shaken so that most of the octanol is thrown off. 10 mL of chilled beer are drawn into the pipet and transferred to a 50 mL centrifuge tube. 1 mL of 3 N HCl is added followed by 20 mL of spectrographic grade iso-octane. The tube is shaken vigorously using a wrist action shaker (though I know one guy who does this determination professionally who shakes by hand) for 15 minutes. If there is a slush in the tube, centrifuge until the iso-octane phase is separate. Pipet a couple mL into a quartz 1 cm cuvet and read against another quartz cuvet filled with iso-octane and a "minute" amount of octanol. The IBUs are 50 times the absorbtion at 275 nm.


Best bet for your home brewing is to pick one of the estimation formulas and note it on all your beers, then you can rank them relatively based on your perception of the IBUs calculated and can better target the amount of bitterness you want in the future.  Since a lot of it depends on your equipment and the alpha acid content of the particular crop of hops.

You might also want to pop over to homebrew.SE and ask if they know of any other test kits or have advice.