Is it okay to feed $I(0)$ variables into the Johansen procedure? I've read three sources that seem to state that this is not what you're supposed to do. However, whenever I've done this, I notice that $\Pi$ is full rank and so it leads me to a VAR and therefore I don't see any problem with this.
I(0) variables is not a theoretical issue. Please see pp 5-6 Hjalmarsson and Österholm (2007) on this.
This is an expected result. The matrix $\Pi$ has full rank, when the process is stationary. Of course Johansen procedure usually requires that the time series should be checked for unit roots first. The null hypothesis is that time series are unit-roots and they are cointegrated. If your variables are $I(0)$ then the first step should eliminate the need to use Johansen's test. I would hesitate to use Johansen's test for testing whether the processes are $I(0)$, since it was not designed to be used as such.