I am not a paladin expert at all but will try to answer this.
1. From what I can determine, you do not have to answer to anyone if you don't want to. This is up to the player and their DM. Your oath is what drives you and you are not bound by anything else including society's laws. I am pretty sure you can be of any alignment in 5e as well.
2. This is also up to the player. Any of these would work.
3. This is a subclass in the DMG, usually players do not pick this one. It gets some pretty interesting stuff though if you were to find a DM who allowed it. With players who stray from their oath, the "serious consequences" I think are usually left up to the DM.
4. This would depend on a lot of things. I do not think an oathbreaker would stand out in any way unless you were specifically told about them. I would think you would only be hostile to them if they stood in the way of your oath.
5. I don't see an issue with this. You can be an atheist paladin in 5e even, although atheism is not recommended in the FR. With the worship of a peaceful deity, you could argue that smiting the enemies would allow for peace to reign or that you are defending those that can't defend themselves.
