If gravitational waves propagate at the speed of light does it mean that they are massless and can not experience time
like a photon?
A bit of oranges and apples (GW are like classical waves; photons are quantum beasts), but yes.
Does it also mean it is no longer true that nothing can travel as fast as light in vacuum?
Yes ... except that GW were expected to travel at c, from circa 1916 ... the only thing that's changed is that GW has now been directly detected.
Are binary black holes caused by binary stars dying at the same time?
This is an active area of research, and (IIRC) there are two pathways to the formation of BBHs (of the stellar mass kind) that are considered most promising. Sorry, but I don't have the papers to hand, and don't want to rely upon memory, but in neither do the two stars in a binary 'die at the same time'.
If binary solar systems are so common why has it taken this long
to discover gravitational waves?
Binaries (ordinary star-ordinary star) are very common; BBHs are likely very rare.
The GW from a BBH inspiral is weak, and the rate at which such events happen not yet well constrained (or even understood); it seems that had GW150914 happened while the original LIGO was 'up', it may have been detected (GW detectors are 'live' much less than 24x365).
How important is the discovery with regard to formulating a theory of quantum gravity
Not at all, at the moment ... except that the detection constrains the mass of any graviton to be incredibly small.
With a few hundred BBHs inspirals observed, plus comparable numbers of BNS (binary neutron star) and BH-NS inspirals, plus detection of such events in the electromagnetic spectrum, it may be possible to put some weak constraints on some QG theories.
Better constraints will come from LISA, or relict GW from inflation (i.e. CMB observations), but I think it's far too soon to do anything more than speculate.
This arXiv preprint may contain a lot that will address at least some of your questions:
Astrophysical Implications of the Binary Black-Hole Merger GW15091