The sentence that is correctly written and punctuated is 4, that is, "to keep his active mind challenged, Roosevelt improved his stamp collection and read books on American history, politics, and biography". This is the case because there is parallelism between the coordinated elements (the verbs), that is, they are both in the past simple tense ("improved" and "read"). What is more, it should be "challenged" instead of "challenging" and the -to infinitive clause of purpose is the right option to tell that to reach his goal (keeping his mind challenged) he improved his stamp collection and so on.