Fuzzy rule based system: Should rules contain all inputs and outputs?
Not really. It is possible to create fuzzy logic model without setting whole ruleset. But there is a risk, that model will perform worse in some unexpected situations which would be handled by missing rules.

However, it is possible to figure out fuzzy rules basing on data. I personally used Wang-Mendel method and it worked fine for me: still 6 inputs and 4 outputs will be many of rules to deduce :(. Wang - Mengel method is described in this paper: http://sipi.usc.edu/~mendel/publications/Wang%26Mendel%20SMC%2011-12-92.pdf