Re: "... a long and wider urethra results in faster flow ..."
It's a bit confusing in the paper - they are taking the length of the urethra to represent the "drop" i.e. pressure difference due to gravity between the header tank (bladder) and outlet, so the longer it is the faster the water comes out at the tap. Of course this neglects the angle of the pipe. They do mention that in analysing the difference between male & female, but it's not clear whether their urethral length data is also gender corrected...
I'm no expert on fluid mechanics, but from the one old textbook I have and a bit of algebra, I come up with a similar 1/6 power law for emptying a volume from a tank with a hole in the side if you scale the lengths/areas/volumes dimensionally according to mass, so I think the rest is more about coming up with an decent estimate for the time, which they get within an order of magnitude.
I take my hat off to them though, from downloading videos of animals pissing off YouTube and holding sawn-off coke bottles under pooches at the local dog park, they managed to get a PNAS paper. Most people would just have ended up in the local paper.
Beer, because I bet that's where it all started.