Assume $\varphi _t $ $C$-dense. If $\nu $ is $\varphi _1 $ invariant is $\nu $ $\varphi $-invariant?
No, cf 
For an Axiom A basic set, C-dense means that the flow is mixing; I think that the terminology comes from the connection with the density of strong stable (contracting) leaves.
Ponce and Varao  recently proved a rigidity result that characterizes when an invariant measure for the time-1 map of a flow will be invariant measure for the flow. Surprisingly, the result is very general and does not require hyperbolic properties for the flow.