Will the lights still be in New year's night (, Monday)?

Yes I believe so

Lights are left on till 12:30 am. So there is half an hour of lighting available on the first of January. For more info, refer to this blog post on the official temple square website. https://www.templesquare.com/christmas/

Sunday the 31st is the last time for the Temple Square lights. They will not be on January 1st.

Yes.