With ECDSA is there a way for the verifier to calculate any properties of $k$?
There are no relation we are currently aware of.
The reason is as follows. The map
$$k \mapsto (k G).x$$ is assumed to be a good pseudo random number generator.
(The NSA infiltration of the Dual EC drgb has nothing to do with that fact).
This basically says that k and r can be seen as independant random variables.