Your option 3 is correct.

When the spell ends, the target can’t move or take actions until after its next turn, as a wave of lethargy sweeps over it.

Not "when the target is no longer under this effect" or "when the target no longer has Haste"; the condition for losing a turn is when the spell ends. So it doesn't matter that there are 2 spells affecting them simultaneously - if a Haste spell ends, you lose a turn.

However, they are still under the effects of a Haste spell.

Until the spell ends, the target’s speed is doubled, it gains a +2 bonus to AC, it has advantage on Dexterity saving throws, and it gains an additional action on each of its turns.

Their speed is still doubled, they still have +2 AC, they still have advantage on Dexterity saving throws, and they still gain an additional action. They can't move, and they can't take actions, but this doesn't prevent them from getting any of those benefits. It just prevents them from using some of them.

This can work

And congratulations on thinking up of a creative use of a buff spell. I like this unconventional application of Haste, as I love that spell myself. Of course, you would probably get more use out of this if you buffed your allies instead. :)

1. Does the casting of haste fail because the target is not willing anymore? Does the target stop being a willing target and the haste spell will end? Does the spell just work as usual?

   • RAI: Yes, haste does end. If you take a look at this Sage Advice, Jeremy Crawford says that a spell which requires a willing target takes the creature's will to help fuel the spell. This implies that their ongoing consent is a requirement to keep the magic alive.

2. Can the caster stop concentrating on the haste spell during the target's turn to make it waste 2 turns?

   • Yes, as you can stop concentrating at any time without expending any resources. This includes at the start of their turn.

   • If the target becomes an invalid target, the spell ends on them. Haste says: "When the spell ends, the target can’t move or take actions until after its next turn, as a wave of lethargy sweeps over it." So they cannot take an action until the end of their next turn.

   • Normally, you only lose 1 turn when Haste ends. This is because it typically ends at the end of your turn (after which it has fulfilled the full 1 minute duration), or you lose concentration of it during some other creature's turn (therefore there is no current turn to lose). But let's say you cast it on yourself, and you lose concentration on it at the start of your turn, you actually become lethargic immediately, and therefore lose that current turn as well as your own next turn.

   • This means, if the target becomes unwilling at the end of their turn, taking into account the answer from #1, they only lose one turn due to the spell ending on them.

Making a Target Willing Against Their Will

1. The most unambiguous and cheapest means to do this is Charm Person. It does not require concentration and lasts for an hour. A creature charmed by this spell regards you as a friendly acquaintance.

   • Depending on the DM, the fact they are friendly to you may not be enough to call them willing targets of the haste spell. In which case, you may have to make a Persuasion check to convince them to let their speed be bolstered by your magic, as you are friendly acquaintances, after all.

2. A more ambiguous -- but still reasonably valid -- method of making a creature willing is the Command spell. If you give it a command to become willing, then it becomes willing for 1 round without concentration. Commands that might accomplish this would be "submit", "surrender", "yield", or "succumb".

3. A high level spell which can definitely do the trick is Mass Suggestion, which is Suggestion but without concentration. So you can suggest to them to "become willing to allow me to enhance your speed with my magic".

4. Another spell which can technically do this, but is not really useful in combat, is Geas. If you command them to "always be willing to let me enhance your speed with my magic", then they must be willing targets of haste or take 5d10 psychic damage. If they become unwilling of their own accord, they take the damage and lose a turn.

It doesn't work for a target under Suggestion, Dominate X, Friends, etc

If you made them willing by using a concentration spell such as Suggestion/Dominate X/Friends, then when you cast haste on them, your first spell will end and there is no time of overlap. This means it is questionable if they are willing in the first place, and hence they may be invalid targets of haste even before haste has been cast on them.

It is completely up to your DM to decide if they still become willing after the two spells (charm and haste) switch up.

If both spells overlap for a short period of time, they lose only 1 turn

Let's say your DM ruled that they remained to be willing targets of haste even after you lost concentration of your Suggestion/Dominate X/Friends spell. So they come under the effects of haste. However, if they immediately become unwilling targets after haste has been cast, haste ends immediately. They lose only their next turn, because it is still your turn now while you are switching the charm and haste spells.