How similar do terms have to be to connect them with 與, 與其，及其？
與： 和，and;
與其： 和他的， and one's;
及其： 还有他的，and also one's;
Back to your examples:


從該日起在此居住與唸書。// fine

2a.該研究美國與其政治方法。// wrong as the reason you said
2b. 該研究美國及其政治方法。// fine, [research on America and its methodology of politics].
2c. 我们很乐意为这所房子及其庭院拍照。 // fine, [take pictures for this house and also its courtyard.]
3a. 辜鴻銘不認可胡適這位新文化的帶頭人與其課程。// wrong, the same reason as 2a
3b. 辜鴻銘不認可胡適這位新文化的帶頭人及其課程。// fine, [disagree/disapprove the leader of ... and also his courses.]