I am working on a problem where response variable is binary and my features are dummy variables. I observed when I include intercept to model all the dummy variables' p-values are equal to 1. When I remove constant p-values seems ok. My question is should not we include intercept when we have dummy variables as our only features? If it is so, what is the reason for that?

Another follow up question is about perfect multi-colinearity, I remember from linear regression, if we do not drop one of our dummy variables we cannot invert the matrix due to multi-colinearity thus we cannot get our coefficients. I know logistic regression coef. found through MLE but I wonder if multi-colinearity still causes different issues in this case?