Do modern Jews still strictly observe those rules set in the Old Testament? Do they still observe those holidays and religious rituals (such as Sabbath day, etc)?
This is a complicated subject that can only be touched on here. Briefly, there is a continuum of ritual observance and belief among Jews both in Israel and in the Diaspora. Some Jews are fully observant and some do not observe anything. Most Jews fall somewhere in the middle, observing some commandments and not observing others for various reasons. Still, most Jews observe certain key traditions to some degree, including circumcision, fasting on the Day of Atonement, not working on Shabbat (though perhaps not keeping to all the traditional restrictions).