I have three children from a previous marriage. I now plan to do an orthodox conversion. What kind of relationship can I have with them if I am an excellent father and they share all weekends with me (by Court order), including Shabbos and most of extracurricular activities???

You will of course be required to observe shabbat (and all other mitzvot such as kashrut). Your children, who are not Jewish, do not observe shabbat. This in itself does not necessitate a conflict. You should speak to a professional in family matters - perhaps a social worker - to discuss how to minimize friction in such a situation.