This is up to him how he wants to roleplay it, and to you as the DM in case it conflicts with his oath
In 5e, paladins do not have to be goody-two-shoes with a fixed code of conduct. Instead of that, a Paladin picks an "Oath" and has to stay true to that Oath. Taking an Oath only happens at third level, and the rules say (PHB, p. 85)
Some characters with this class don’t consider themselves true paladins until
they have reached 3rd level and made this oath. For others, the actual swearing of the oath is a formality, an official stamp on what has always been true in the paladin’s heart.
So until they have taken the Oath, it is entirely up to the player how he sees his paladin. If the does not consider himself a true paladin until then, that is fine, and he still will have access to his class features.
Once he has taken an Oath, there will be some tenets that he has to adhere to and if this behavior violates them will depend on the Oath he has taken. For example, the Oath of Devotion include a tenet called Compassion:
Compassion. Aid others, protect the weak, and punish those who threaten them. Show mercy to your foes, but temper it with wisdom.
Going crazy and just slaughtering hapless foes would violate this tenet.
What are the consequences if he does violate his oath? The rules (inset Box Breaking Your Oath on p. 85 PHB) have this to say about breaking the Oath:
A paladin tries to hold to the highest standards of conduct, but even the most virtuous paladin is fallible. Sometimes the right path proves too demanding, sometimes a situation calls for the lesser of two evils, and sometimes the heat of emotion causes a paladin to transgress his or her oath.
A paladin who has broken a vow typically seeks absolution from a cleric who shares his or her faith or from another paladin of the same order. The paladin might spend an allnight vigil in prayer as a sign of penitence, or undertake a fast or similar act of self-denial. After a rite of confession and forgiveness, the paladin starts fresh.
If a paladin wilfully violates his or her oath and shows no
sign of repentance, the consequences can be more serious.
At the DM's discretion, an impenitent paladin might be
forced to abandon this class and adopt another, or perhaps
to take the Oathbreaker paladin option that appears in the
Dungeon Master's Guide.
Here, the way this is played is that the paladin character is losing it, Jekyll-and-Hide style, so it may not be exactly wilfully violating the Oath in a specific instance. He is not really in control of his actions. But he still is breaking the Oath, so he should repent after doing it. This may not be that practical -- there is a lot of combat in 5e, and he would pretty constantly have to be full of remorse and repentance for what he is doing. But maybe it works for some time. He constantly spends his nights in atoning prayer, fasting etc. to make up for it, and visits the temple as much has he can.
It is your call as a DM if you want him to change his class eventually because in the last consequence, if what he does is not in line with the oath, he is wilfully breaking it if he knows this is going to happen again and again and opting to continue like that.
While the rules say that this is the DM's discretion, a good approach can be to discuss with the player out of game how he envisions this to work and this character to develop, collaborating on the character concept. This can help generating buy-in from the player if you ultimately decide the character can not stay a paladin long term under their oath with this behaviour, or it can help to pick an oath that does not conflict strongly with it.
In any case, there are no rules that say he loses access to his class powers when he violated his Oath, until he stops being a paladin altogether and maybe ends up as an Oathbreaker.
So, to your questions in particular:
- This is the players decision on how he role-plays his character.
- Nothing happens, he will have access to his Paladin abilities.
- This does not apply; either he has his abilities, or he needs to stop being a paladin altogether.
P.S. A brief history of the Paladin class
When the Paladin was originally introduced, it was an extremely powerful class and an exclusive pleasure. Access to play a paladin was gated by minimum ability score requirements that were extremely hard to hit using the standard methods of score generation -- about 1.85% in OD&D with expansions, and 0.1% in AD&D 1e.
These paladins had to follow strict behavioural constraints.
unlike normal fighters, all paladins must begin as lawful good in alignment (q.v.) and always remain lawful good or absolutely lose all of the special powers which are given to them.
If a paladin should ever knowingly and willingly perform on evil act, he or she loses the status of paladinhood immediately and irrevocably. All benefits are then lost, and no deed or magic can restore the character to paladinhood
Harsh. Long-time players may have this at the back of their minds, when they think of paladins. The reason for this was to balance the wide range of extra powers that paladins got. Yes, you were super powerful, but to make up for it, you had to behave. 1e DMG, p. 24:
It is of utmost importance to keep rigid control of alignment behavior with respect to such characters as serve deities who will accept only certain alignments, those who are paladins, those with evil familiars, and so on. Part of the role they have accepted requires a set behavior mode, and its benefits are balanced by this. Therefore, failure to demand strict adherence to alignment behavior is to allow a game abuse.
The game has come a long way since those days, and today in 5e, the different classes are much more balanced in power. The paladin is no longer a much stronger class mechanically than a fighter or cleric. Because of this, the need to strictly enforce behavioural constraints is not really there any more, and there is no game balance need to deprive the paladin of their powers if they fail to adhere to them. This opens up a lot more different ways to portray or imagine paladins.