You cannot take an action such as the Cast A Spell action or the Attack action (and thus cannot grapple/shove) but besides that the choice is up to you
TL;DR:
Grapple/Shove and casting a spell require specific actions, (the Attack action and the Cast A Spell action respectively) and so we cannot use them when being compelled by crown of madness
The spell simply says we must make a melee attack, it does not specify any further what kind of attack this needs to be but we can look to similar features and their related questions as guidance.
Opportunity attacks, the Hunter Ranger's Whirlwind Attack feature, and the crown of madness spell all use the same wording so we can use previous rulings on the related features to determine how crown of madness should work:
To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature...
You can use your action to make a melee attack against any number of creatures within 5 feet of you...
The charmed target must use its action before moving on each of its turns to make a melee attack...
The answers to the related questions "Can you use a melee spell attack with the Hunter Ranger's Whirlwind Attack?", "Can you grapple/shove when affected by the Crown of Madness spell?", and "Can I use a spell to make an Opportunity Attack?" explain that melee spell attacks and the special melee attacks of grapple/shove are not options for us. Because there are no further restrictions on what kinds of attack we can make all other melee attacks are available to use, in particular these are unarmed strikes, and melee ranged attacks that use weapons (even improvised ones).
The reason we cannot grapple/shove is because these require us to take the Attack action as the Sage Advice Compendium shows:
Grappling and shoving are special melee attacks that require the Attack action (PH, 195).
This is also shown in the linked question above about crown of madness where it is explained that you cannot grapple/shove when under the effects of this spell. This is further supported by the sections on grapple/shove:
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple...
Using the Attack action, you can make a special melee attack to shove a creature...
The reason we cannot make a melee spell attack is because those require the Cast A Spell action as the Sage Advice Compendium explains:
Q. Can you use a melee spell attack to make an opportunity attack?
A. You can’t if the spell attack is created by casting a spell. When a creature triggers an opportunity attack from you, you can use your reaction to make a melee attack against it. The opportunity attack doesn’t suddenly give you the ability to cast a spell, such as shocking grasp.
Each spell has a casting time. A game feature, such as an opportunity attack, doesn’t let you bypass that casting time, unless the feature says otherwise.
This is supported by the other two linked questions above "Can you use a melee spell attack with the Hunter Ranger's Whirlwind Attack?" and "Can I use a spell to make an Opportunity Attack?".
Opportunity attacks and Whirlwind Attack also do not allow us to make melee spell attacks, and as they have the same wording as crown of madness, you cannot make a melee spell attack while under its effects either.
None of these feature specify that we can bypass the casting time requirements of a melee spell attack and so we cannot do this.
Thus we have removed both grapple/shove and melee spell attacks from out list of available options. The only other melee attacks are melee weapon attacks (unarmed strikes, and making a melee attack with a weapon, even an improvised one).
The spell never limits any further what kind of attack you have to make, it doesn't say it has to use a weapon, or that it cannot be improvised and so you can choose to make any kind of melee weapon attack.
That said, this does make the spell weaker; It already requires your action each turn to continue, and the creature makes another save at the end of each of their turns, so it is not likely to last very long.
After those setbacks, allowing creatures to deal only 1 + Strength modifier damage, which is quite pitiful on many enemies, weakens the spell. Its only use would be to prevent a target from using its action in some other way, and though this is quite strong it makes the controlling portion of the spell have nearly no benefit.
Because of this it would be well within a GM's purview to require that the attack actually uses a weapon in its intended way thereby dealing a sizable amount of damage.
A note: In the question "Can you use a melee spell attack with the Hunter Ranger's Whirlwind Attack?" there is a great amount of discussion in the comments and answers about what you can actually do when something allows you to "make a melee attack", there doesn't seem to be any sort of consensus here besides the top answer's flat "no" and so I've used this as a correct answer.
Yes
There are two people who can know whether the attack hits; the player or their character. In both cases the answer is the same.
The Player
In every game that I've played, players are not supposed to have knowledge of a creature's statistics, including to-hit modifiers. Additionally, they do not know all the different spell and feature effects that may change the attack roll before or after it is made. They may be able to deduce the these values through observation, however as they are not the DM, players cannot have absolute knowledge of the final value of a roll.
Therefore, a player can never know for certain whether the attack hits or misses until the DM declares whether or not the attack hit. So, as long as you use the feature before the hit is declared, you should be able to use this feature.
The Character
Despite the usage of the word "you", descriptions for spells are typically written as a description of what the character can do, not you as the player. I believe that the following is the most likely RAI interpretation:
The character has no knowledge of a creature's statistics, including a creature's to-hit modifier and their own AC. These exist purely as a way of indicating how good a creature is at hitting things, and as a measure of how difficult it is to hit a creature, respectively, from a gameplay perspective. They have no real in-game counterpart. Therefore, you seeing the value of the roll does not impart on your character any knowledge of whether the attack hits or misses them. As such you can use this feature after seeing the roll.
This question contains further elaboration on this topic.
Best Answer
Using your action extends the duration on both creatures
I'm basing this answer on the way other spells behave when being split or twinned. References to a single target apply to both.
This is for Hold Person, as an example (emphasis mine):
With the spell twinned, now target refers to both creatures. I see no reason for the action to continue the spell to not apply to both as well.
Furthermore, here's a tweet by Mike Mearls in response to a similar question:
That tells me, the Split spell continues to act as close as it did before but with the added second target. You are not concentrating on two spells, and you don't need to spend an action on each creature; one action works for both.
See Witch Bolt for a similar case (PHB, p. 289; emphasis mine):
If twinned, your action lets you damage both the new targets. The wording is a bit different but wording of spells isn't always 100% consistent from what I've seen. I think it's very likely Crown of Madness is intended to work the same way.
As for the other questions:
No, you can't. If we treat it like any other spell with multiple targets (say, for example, an upcast Hold Person), by RAW, you can drop concentration at will, ending the whole spell, but there's nothing that lets you end only part of the spell.
This requires some development.
First, I think you were asking mainly in the case that you couldn't keep both crowns with one action, or if you could choose to make one stop at will while keeping the other. Since neither of those is the case, that part of the question becomes redundant.
However, I think there is still a case where it's worthwhile to consider what happens. It's quite a common situation, actually, because that situation is, what happens when one of the targets makes the save?
If we compare the wording of Hold Person above to Crown of Madness (PHB, p. 229; emphasis mine):
It would seem there's a difference in the way the spells stop. A creature passing the saving throw of Hold Person ends it on itself, while a creature saving for Crown of Madness ends the whole spell. That would be in line with this answer to a similar question.
Now, my contention with this is that single-target spells aren't written to reflect their behavior when applied to multiple targets. Even Hold Person speaks about a single target (emphasis mine):
When it should read something like (emphasis mine):
So, the difference between Hold Person's stopping condition and Crown of Madness' can be chalked up to the fact that Hold Person can be upcast to affect multiple creatures, so that situation was taken into account for a very in-game relevant aspect of the spell. But the rest of the wording of the spell is left simple to fit the single target use of the spell.
Answering the question: By RAW... maybe?.
My position is that RAW is not written for multicasting these single target spells, so modifications should be made to make the spells make sense. I think it's reasonable to modify the stopping clause of the spell as well to make it more like Hold Person's. Now, someone could take the stance that only the minimum number of modifications should be made for the spell to make sense and still abide by RAW as much as possible and the stopping clause doesn't need to be changed to work in a multicasting scenario. And that is a reasonable stance too.
Still, by my interpretation of the rules, definitely.
Replacing each instance of "target" with "each target" might work, "any target" may be even clearer.
I would also replace: "On a success, the spell ends" with "On a success, the spell ends, for that target".
Note that this doesn't make the spell particularly clearer for a normal reader without a multicast feature, which is likely be the reason why the spells aren't written that way to begin with.