How do you compare unpaired biological replicates?
Calculating fold change is pretty straightforward but you cannot do one on one comparison because they (control and treated) are not really paired (same sample). That's why IPA gives you six results (FC for each pair). However, you can calculate fold change between two genes because they are from the same sample. Note that fold changes are quite dangerous and can lead to misinterpretation of data (for example 1/2 would be same as 100/200). You should use them cautiously.

I think, in this case, you can simply apply a t-test. This will tell you if your two groups i.e. treated and control are significantly different from each other. You would not be calculating fold changes here. If you had more number of groups then you can go for one-way ANOVA.

I am not sure what output you really want after the analysis but I think the main goal is to say that control is different from treated.

Since you have multiple genes (variables) you can either go for a multivariate t-test, which will tell you if the control is overall different from treated, or many univariate tests which will tell you how each gene differs between the two groups. There are some methods such as Bonferroni correction which can be used to improve the prediction of multivariate tests.

When you have established that a certain gene is differentially expressed, using t-test, you can report the fold change of the averages.