I would think regression or ANOVA would work better than t test, but it seems that you are simulating the data points rather than having real data. How will you know if what you simulate will match the reality you are trying to mimic?
You have multiple confounds and need a design to address this. I believe, it has been many years, that methods such as Latin Squares permit you to test only a portion of the potential confounding features rather than all levels of each. You might look this up in a book on designs. Exploratory factor analysis will identify common factors in your data. I don't understand how they would test if certain factors had more influence on the dependent variable you are interested in.
It is entirely possible I simply misunderstand what you are looking for. I assume you have many variations of conditions and you want to find out which level of which has the greatest impact on some dependent variable. If so, you have to be able to gather data on the levels of the iv and the levels of the dv associated with it. I have never seen EFA used for that.