In classical geometry the calculation of the Chern classes of a vector bundle using a connection is independent of the choice of connection. Does any such result hold for projective modules in noncommutative geometry?

Yes. You can see the construction in detail, for example, in Max Karoubi's ‘Homologie cyclique et $K$theorie’ (Asterisque 149, SMF; you can get this from his web page), where he constructs the Chern classes $K_0(A)\to H(A)$ using connections much à ChernWeyl (Here $H(A)$ is the noncommutative de Rham theory, or one of the various cyclic homologies of $A$) He also constructs higher Chern classes on the higher algebraic $K$theory by a similar procedure. The book by Loday on cyclic homology also covers this. 

