on their next attack roll equal to their current hit points + their number of current spellslots, as well as the +6 bonus, and then only on their next successful attack roll.
A:
A paladin can use Smite Evil on an undead creature once per day (PHB 115). This is a supernatural ability.
A paladin can make an attack at higher initiative if they use their Action Surge twice (PHB 117). At higher initiative, a paladin will need to make a single attack roll to damage the target. Thus, the paladin will only make one attack roll against the creature that day.
If a paladin's current hit points are 13 + number of current spell slots, then they make one successful melee attack with two-weapon fighting (PHB 195). If they succeed this attack, they will deal half the damage from both weapons and they will be exhausted for a round, but they are not exhausted as a result of using an action surge (PHB 117). Thus they can use that bonus action to make a second attack with the same weapon.
If the creature has the same constitution score, use of two-weapon fighting on the creature's turn does not cause disadvantage on attack rolls against the creature (PHB 195).
Note: I would not recommend this.
The advantage gained from one attack not exhausting the creature is better than the advantage gained from two attacks. However, the disadvantage is that each additional attack on the creature’s turn results in a full attack rather than an extra attack. However, this is not a big deal if the creature has no opportunity attacks and is immune to two-weapon fighting. This ability could help get rid of a creature quickly but will take a lot of resources (attack rolls).
A:
No.
A paladin cannot cast a spell during their own turn, for any reason.
The only exception to this is that a Paladin can choose to make an attack with Smite Evil instead of making a spellcasting ability check. The Paladin can only do this once per day, but that's still an attack, not a spell. So no, a Paladin cannot choose to make two attacks, nor can they use their action surge more than once per round.
You are limited to a certain number of spells known. At 2nd level, you have two 1st level spells righter, and you learn one additional spell of your choice from any school of magic.
You know two cantrips of your choice from any school of magic
If you want to apply multiple attack rolls, then use multiattack or some other method.
While you are conscious and wearing your holy armor,
your Strength and Dexterity scores each increase by 2.
... and ...
You have advantage on saving throws against spells and other magical effects.
However, at that point you are no longer a paladin.
A:
RAW, probably not
Two main pieces of evidence to start with:
Once per day, a paladin can choose to use their Channel Divinity to attack with an evil-aligned weapon, rather than an attack roll with a divine spell (see "Smite Evil" sidebar on page 11 of the PHB).
There's no rule about attack rolls being only available once per round, so unless we give paladins extra attacks on their turn, that's a reasonable interpretation that we're giving a bit of power here.
That said, you've correctly noted that any additional attacks still involve taking your action and using your action, which means that you aren't taking multiple attacks in a single round. See this question for more about that.
Paladins do not have spells
They're a class that is meant to fill in the gaps in a character's arsenal that would otherwise leave them without access to magical powers. They are not granted access to a full spell list, just a selection of them, and access to one extra spell per day.
Since they can't cast spells, you won't get the advantage from using both abilities, since they don't work that way.
This isn't my interpretation, and I think it's an interesting interpretation of the rules. However, it doesn't really make sense to me.
A:
No, they can't
Paladins are supernatural creatures
In the PHB 117, under the section "Action Surge (pg. 117), there is no section allowing "extra attacks" on an attack made using your Action surge. You are limited by the number of attacks available and the number of turns you are given.
So even if you were to consider the attack you get when using action surge to be a bonus action attack, that still means that all of your attacks from an action surge will count as one action and have one standard attack roll.
Bonus Actions
Most class features that grant you additional actions describe what you can do in addition to your regular action.
Your class might give you an action such as dash, dodge, or disengage, and your character might have a special ability such as improved uncanny dodge that allows you to dash or disengage as a bonus action. You can also take the Ready action during your turn to give yourself extra actions for later in the round.
The rules of D&D don't have a set limit on how many attacks you can have when using an action surge. Some options allow you to make extra attacks, while other class features have attack rolls that are always at 1/turn. This means that a paladin is not allowed to have more than one attack from any given action surge.
However, it would be an interesting tactical choice for them to use an action surge to get a bonus attack and then use a single-use spell like Shocking Grasp or similar to deal damage on their next turn.
You've answered that answer yourself!
No, they can't.
If a creature's hitpoints equal its constitution mod (excluding abilities) then it is able to use its full action in combat. If a character is wielding two one-handed weapons then they can make one attack on each turn, as long as you are wielding that weapon.
So when a paladin uses a turn with an action surge, they only have one thing to choose between: Cast Smite Evil (their first action of the turn), or attacking the enemy (their second action of the turn). Since they cannot take an attack action, they cannot use their action surge.
Furthermore, in the paladin's own turn, since he has already used his reaction to cast smite evil, he has nothing to choose between attacking or smite. In such a situation, he cannot cast smite again. His ability to cast Smite Evile only occurs once per round.
You are able to attack a number of times during a round equal to your Constitution modifier + your proficiency bonus. (PHB, 196).
With that many attacks available, the paladin has an easier time attacking the enemy first with a bonus action, and then potentially attacking a second time as their reaction.
Re: 'the target number for an attack made as part of a single attack action', which is 8(PHB, Page 193, Attack Rolls), a paladin cannot make an attack action without spending an action surge and two regular attacks.
However, a paladin's smite evil does not provoke an attack of opportunity as his smite evil doesn't allow him to make a melee attack. This means that he can use his action surge on his smite, allowing him to make multiple attacks with smite when he reaches level 5. However, only one attack for each action surge spent (3 actions spent = 6 attacks).
Also, when using the
