The in sample Granger Causality test is essentially a significant test. If lags of one variable significantly affect current value of another variable, then one say this variable granger cause the other variable. So you need to focus on the problem you analyze. Of course, using some notation may make your analysis looks "advanced", but it's not the most important thing.
Adding other variables may change the result, even the sign of the estimated coefficient may change. This may due to the endogeneity problem, you may read some elementary books for this, for example Woodridge's book. When the added variables are independent of the original vairable, the result may not change.
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