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 < $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.
From Ray Daniels' Designing Great Beers:
To really know the level of bitterness present in a beer, you must have a laboratory analysis performed.
For home brewing purposes, the best you can usually do is to plug the alpha acid values of the hops you're going to use into an IBU calculator such as the one from Brewer's Friend or using Beersmith to calculate the bitterness. (The alpha acid values can be found on the hops' packaging.)
It's worthwhile to not get too wrapped up in IBU. From Daniels:
Using all these techniques … the big guys only control their bitterness to within plus or minus 2 IBU! Given that the average level of bitterness might be something like 15 IBUs, that represents an allowable variation of 13 percent in the bitterness of the product. The reason [major producers] don't sweat over the IBU inside that range is because the human palate can't detect the difference. Studies have shown that the detection threshold for bitterness is about 5 IBU. And that is for beers in the 10 to 15 IBU range. Among home and craft brewers, the sensitivity is usually less because we are used to more bitter beers.