Regarding socio-sexual signaling, the majority of pterosaurs had either a head-crest or a tail-vane.
_Rhamphorhynchus_ had a tail-vane, but no head-crest; as did _Sordes_. _Pterorhynchus_ had both. Most short-tailed pterosaurs had a head-crest of some kind. One notable exception is the anurognathids - but is this because they might have been crepuscular/nocturnal, and visual displays weren't much use in the dark?
On the rigid tail in long-tailed pterosaurs, as Habib & Hone note:
"This functional explanation is consistent with striking similarities between the tails of dromaeosaurids and rhamphorhynchids, including the relative dorsoventral rigidity of the tail (provided by caudal rods) and the comparative lateral mobility of the tail (Persons & Currie, 2013)."
Persons & Currie (2013) devoted a whole paper to the uncanny resemblance between the tails of rhamphorhynchids and dromaeosaurids, with their caudal rods that constrained up-and-down motion of the tail. Persons & Currie came up with some heterodox explanations (such as derived dromaeosaurids being secondarily flightless, and running like Groucho Marx). Back in 1968, Ostrom proposed that the stiffened tail of _Deinonychus_ worked as a dynamic stabilizer during predatory attacks, such as when leaping against larger prey.
Among long-tailed pterosaurs, not all have the tail reinforced by caudal rods, and even among basal pterosaurs this feature varies - so it wasn't essential to early flight in pterosaurs. Maybe the caudal rods allowed the long tail to be held off the ground during leaping, and/or prevented it from contacting the ground during ground-to-air take-offs in pterosaurs and volant dromaeosaurids like _Microraptor_.