Does social standing have a place in your game? Yes it does factor in a PCs interaction with various levels of nobility/royalty should they have some social standing of significance.
When you make up a background for your character how much depth do you put into it? Personally I put as much detail into it as the DM's game will allow. In a setting I am familiar with never less than a full page of background. If it is a DMs homebrew and outside of the typical norm then how much backstory I have will depend upon how much time the DM has to devote to enlightening me to the point that I can write a backstory that fits the world.
Do you have rules set in place for backgrounds?
I have never felt the need to set anything in writing as it is all very subjective depending upon what the player is really looking for and how well that fits into the campaign.
Does your DM give you bonus skill points of feats to help flesh out a background?
No not that I am aware of. I would probably be in favor of allowing some small advantage in terms of starting equipment and relations coupled with the understanding that there are certian responsibilities and expectations that go hand in hand with such a background.
Will your DM allow you to play a prince?
I never asked so I cannot answer, however in my game I would give due consideration to such a chatacter proposal. It would require some extensive ground rules/negotiations to actually make it happen.