That's awesome that you are getting the dye to lase! I wouldn't have expected that to work so well in your setup.
As for the broad spectral width of your output, this is typical of organic dyes in a free running setup like you have. The large bandwidth actually becomes very useful in certain applications. If you throw a prism or gratings in your lasing cavity, you can dramatically shrink the spectral width of the output, as well as tune the emission wavelength. According to wikipedia, the dye Rhodamine 6G can be made to lase from 570nm all the way to 660nm using methods like this!
The coolest effect from a medium with a large spectral bandwidth is that you can use it to make incredibly short pulses. This is an effect of Heisenberg's uncertainty principle (in time and energy) where a large uncertainty in energy (a wide range of frequencies) corresponds to a well defined time interval (a very short pulse). IIRC the first lasers producing femtosecond scale pulses were organic dye lasers. A femtosecond is INCREDIBLY short, 1fs= 0.000000000000001s
And for the record, the time-energy uncertainty isn't fully accepted in physics as of now, but the same effect in fast-pulsed lasers can be described with fourier transforms and classical electromagnetic waves.
