do we make personal requests on Rosh Hashona ? is there allowed personal requests and not allowed? what about other special days?

Shalom, Thank you for your question. Yes, it is appropriate to add personal requests to one's silent Amidah (just before taking three steps backwards at the end).This is true on both Rosh HaShona and Yom Kippur (even when they fall on Shabat). However we refrain from such requests on a regular Shabbat. (Hilchot Shlomo, Rosh HaShona, 1, 15) Blessings.