From the Core rules pdf (pg 60):
Detect Evil (Sp): At will, a paladin can use detect evil, as the
spell. A paladin can, as a move action, concentrate on a single item
or individual within 60 feet and determine if it is evil, learning the
strength of its aura as if having studied it for 3 rounds. While
focusing on one individual or object, the paladin does not detect evil
in any other object or individual within range.
In the Detect Evil spell it lists a Cone shaped area effect. So my reading of that is that the paladin can do either but not both at the same time. For example a paladin could be talking to three npc's and use the area effect to see if any of them are evil and then after that move is completed on their next movement action focus on one individual to get a full aura reading but then he would stop detecting evil in the area and only focus on that one npc.
Of course it doesn't seem totally clear cut from the rules explanation so individual players/DM may interpret the rule differently.
None of the Oaths really do more damage than the others as such. The only Oath feature that directly impacts your damage is the Oath of Devotion's level 20 ability, Holy Nimbus - it deals 10 radiant damage to any enemies within 60 feet of you. So, to answer your question directly, Oath of Devotion does the most damage.
This is, obviously, a fairly ingenuous answer. The real answer is that you can't look purely for damage.
For example, both the Devotion and Vengeance Oaths have features that increase the accuracy of your attacks, which in the long-term, will increase your damage.
On the other hand, the Oath of the Ancients has an ability that makes you and your allies much tankier and harder to kill. How does that boost damage? Well, you can't deal damage if you're dead, so anything that keeps you alive is also indirectly boosting your damage.
Then there's things like spells. The Oath of Vengeance gives you access to hunter's mark, which potentially lets you add 1d6 damage to every attack you make. But in practice, it's not quite so simple - it requires concentration, and you also have to move it around to make sure it's on the enemy you're attacking. And, obviously, it requires you to make a lot of attacks to maximize the damage it gives.
As a counterexample, the Oath of the Crown gives access to spirit guardians, which deals damage to enemies who come near you - under the right circumstances, this will vastly outdamage hunter's mark. But in the wrong circumstances, it might be completely useless.
So, in the end, there's no easy answer - you'll have to look at all the Oaths and determine which one you think will work best for you.
Best Answer
A Paladin's Code always applies, no matter the enemy.
There are many, many arguments about alignment and Paladin's Code enforcement around the internet. The angle that I'm arguing here is not the only valid angle. Others, like the one given in @JackLesnie's answer, are also totally valid depending on your group.
The Code of Conduct Paladin feature says the following:
By a strict RAW reading, the text is pretty clear. If a Paladin lies, cheats, or uses poison, they lose all class abilities. Nothing in the text mentions context or mitigating circumstances, even when you're fighting evil outsiders.
I suspect that this is mainly because Paladins are supposed to not just be Lawful Good, but Lawful Good and then some. A Lawful Good rogue could totally backstab, betray, poison, or lie to a demon without a twinge of guilt or alignment shift, since they are just using the tools at hand to fight evil. A Paladin, however, has sworn to be above that kind of combat, and won't stoop to the demon's level by using dishonorable tactics.
To directly answer your list of questions:
Can a paladin Backstab a demon? No.
Can a paladin lie to a demon? No.
Can a paladin steal from a demon? No.
Must a paladin offer mercy to a demon?
If a demon begs for a fair trial or for mercy (even if the paladin knows the demon will try to escape), must he take the demon as a prisoner and treat it with the due respect to a prisoner? Is "Summary execution" a fair trial enough?
The last two deserve more careful analysis. The Code of Conduct says that the Paladin has to "help those in need (provided they do not use the help for evil or chaotic ends)". If a demon asks for mercy and the Paladin expects the demon to just use that mercy as a way to escape and survive, then they would be using the Paladin's help for evil ends, and so the Paladin is allowed to ignore their plea.
Similarly, if a demon asks for a "fair trial"; Nothing in the Code explicitly states that a Paladin must accept a surrender. Unless there's an overarching legal structure that the Paladin must follow that requires something else, nothing in the Code of Conduct says that the Paladin needs to give a demon anything but a summary execution. After all, the Code says that they have to "punish those who harm or threaten innocents", and killing a demon does just that.