Should one examine the cross-correlation plot to rule out performing Granger causality test? Should one look at cross-correlation plot before performing Granger causality test to avoid type I errors?
If we can't find any dependence between two series from the cross-correlation plot, then should we always not perform a Granger causality test?

No.
First, if you are really concerned about type I error rate, use a lower significance level in your test. That is why we have the "significance level knob" built into statistical tests!
Second, it seems you consider using the cross-correlation (CCF) plot as a substitute for the Granger-causality test to avoid doing the actual test. This is not a perfect substitute, but they are related, so I get the intuition. However, you would not gain anything this way. If one method (the CCF) shows absence of a certain relationship, the other method (Granger causality test) would tend to do the same – to the extent one can substitute for the other. So if you see absence of certain correlations, you will tend to find absence of Granger causality, and that would not systematically affect your type I error rate.
