(1) Dua can be made any, time there is no harm. However, in some particular time making dua and offering nafl salah has more virtue and it brings more reward. The best time is that of tahajjud. Sitting at the same place after Fajr salah till the sun rises high and getting oneself involved in dua and tashbih is also one of the best acts.
(2) Yes, the nikah is not valid without the presence of witnesses. At least two Muslim witnesses should witness the event and hear the ijab and qubul (formal exchange of words) that solemnize the nikah.
(3) No, (nikah will not be valid by only making Allah witness).
Allah (Subhana Wa Ta’ala) knows Best
Darul Uloom Deoband