Using machine learning specifically for feature analysis, not predictions
You don't need the linear regression to understand the effect of features in your random forest, you're better off looking at the partial dependence plots directly, this what you get when you hold all the variables fixed, and you vary one at a time. You can plot these using sklearn.ensemble.partial_depence.plot_partial_dependence. Take a look at the documentation for an example of how to use it.

Another type of model that can be useful for exploratory data analysis is a DecisionTreeClassifier, you can produce a graphical representation of this using export_graphviz