When the type of test does not fit the research purpose My research aims to find whether the type of images that my participants were exposed to affected their altruistic behaviours (yes, no) + whether their gender had anything to do with it.
My dilemma is that I'm forced to do a binary logistic regression as my DV is dichotomous. But it does not seem fitting for my research at all. Logistic regression just predicts which IV explains the DV the best and it looks at the whole model and some odds ratios (extremely confusing). I'm not interested in this. In addition, adding several variables in the logistic model changes the significance of the previous variables! Some stop being significant depending on whether others are there too:(
What I need is what anova does. I would like to know if there is a significant difference between my factors in affecting my DV; whether there is an interaction between my factors and "the degree to which one factor is differentially effective at each level of a second factor" (simple main effects).

First review your understanding of ANOVA & its relation to (ordinary least-squares) regression. "In addition, adding several variables in the logistic model changes the significance of the previous variables! Some stop being significant depending on whether others are there too:("—this isn't peculiar to logistic regression. See e.g. How can adding a 2nd IV make the 1st IV significant?.
Second, with logistic regression (& other generalized linear models) you can calculate the equivalent of an ANOVA table with tests carried out using the asymptotic distribution of likelihood ratios according to Wilks' Theorem. See What are the differences between ANOVAs and GLMs?. It's sometimes called analysis of deviance.
Third, point & interval estimates of model parameters are much more informative than declarations of significance/insignificance. Can't your questions be answered quantitatively rather than dichotomously? See e.g. Why is "statistically significant" not enough?. If odds ratios are new to you see Help me understand adjusted odds ratio in logistic regression for a start.
