Q: Do objects exist independently? For example, if one is not seeing the moon then does moon exist or not?
A: From the perspective of absolute reality (paramArtha), of course there is no problem; no question or answer! There is only Brahman; no creation and no objects. But I assume that your question relates to empirical reality (vyavahAra). Here, Advaita teaches that Ishvara governs the ‘creation’, setting and maintaining the physical laws that apply to the universe and the karmic laws that apply to the jIva-s. It is only some post-Shankara philosophers who try to make out that there is only one jIva so that, as soon as this jIva is enlightened, the apparent creation comes to an end. You can read all about the ‘world disappearing on enlightenment’ in the seemingly endless discussions we had on that topic beginning in 2020 (I think).
So, as regards your specific question, objects continue to exist when you go out of the room (for example). Otherwise, other jIva-s would not be able to enjoy them! Suppose that you go outside at night with a friend and both look at the moon. And suppose that you turn away but your friend doesn’t. If the moon ceased to exist, so would your friend (who is also an object at the gross level).
Advaita is not subjective idealism. Objects are not ‘in the mind’ (although the names and forms that we give them ARE in the mind – hence we can see a rope as a snake). But the moon is not ‘real’ in an Advaitic sense; it is mithyA. The story of the sage and the wild elephant is relevant here. A seeker saw his guru run away when a rogue elephant charged. Afterwards he asked why his teacher had run when he would say that the elephant is mithyA. The teacher replied that the ‘running away’ was also mithyA. (At least, that is how I recall the story.)