Light fixtures (as attached) have 2 and 2 tubes in series. If one fails, the other stops working as well. If one blinks, the other blinks too, but in a different rate.
Are there any other benefits except the lower price thanks to only 1 ballast per 2 tubes? What are the cons?
The older building where I currently work uses these with an interesting mix of tubes - like Osram and Philips in series, or GE and Osram etc. Is this ok (the tubes are so standardised that it makes no difference)? And isn't the fact that they probably mix an old tube from the original pair with a new tube an even bigger problem?

Given these ballasts dissipate almost 10W, it is more a question of either 92W (4x18W+2x10W) vs 112W (4x18W+4x10W) power input for the wole 4xF18T8 fixture.
It is really the ballast efficiency, what dictates that connection. The ballast cost is then "on second place"...

Good point. But are the losses really independent on the ballast size? Isn't it more like 2x10W vs 4x7W or so?

Well yes, but the ballast "size" counts on the product of the voltage across the ballast and the current. And there the 1x18W operates ar higher voltage drop (223V) than the 2x18W (203V), so actually the single lamp type is the bigger one here.

Plus the double lamp uses to be designed for a bit higher current (0.43A) for the single F36W operating at slightly higher ballast voltage drop (with the same dissipation as the 1x18W, so has lower resistance), so at the lower drop lower current it dissipates even less than would correspond to their voltage drop.