On the Abishai's first turn it will cast darkness heavily obscuring an area from the players, presumably with them inside it. The Abishai will then attempt to stay within the darkness fighting the players inside of it using its action to attack and then its bonus action to hide.
Yes, this is a sound strategy on its part. The only thing I would add is that the Abishai will first attack (action), then Hide (bonus action), then move. Moving out of the reach of its opponents will not provoke an opportunity attack, because the party cannot see it, but it will make it more difficult for them to guess where it is in the darkness.
The Abishai rolls a hide check against the players passive perception? The abishai rolls against an active perception? Do players need to use action to find the abishai or is it given for free?
On its own turn, the Abishai Hides. Its attempt is immediately compared against the Passive Perception of each member of the party, as explained in the Hiding sidebar of the PHB section on Using Ability Scores (Dexterity):
Passive Perception. When you hide, there's a chance someone will notice you even if they aren't searching. To determine whether such a creature notices you, the DM compares your Dexterity (Stealth) check with that creature's passive Wisdom (Perception) score
Depending on its roll, each of the party members might or might not still know where the abishai is, largely based on its noise.
Then, on each of the players' turns, any character that did not already know the location of the Abishai through their Passive Perception, but who wanted to know enough to use their action for the attempt, could attempt to determine its location by using the Search Action to make an active Perception Check, as described in the PHB section on Actions in Combat (Search):
When you take the Search action, you devote your attention to finding something. Depending on the nature of your search, the DM might have you make a Wisdom (Perception) check or an Intelligence (Investigation) check.
Basically if their Passive Perception was not enough to beat the Abishai's Stealth, they can spend an action to attempt an active roll that is higher. (Note that the DM may permit an active roll, but does not have to).
If we assume the players cannot find the abishai are they allowed to attempt to make attacks against it and hope to hit? Would those attacks simply be at disadvantage or would they automatically fail since they have no idea where the creature is?
Characters are permitted to attempt attacks, but they must first guess the square that the abishai is in. If they are incorrect they automatically fail. If they are correct they can attack with disadvantage. This is explained in the PHB section on Unseen Attackers and Targets:
Combatants often try to escape their foes' notice by hiding, casting the invisibility spell, or lurking in darkness.
When you attack a target that you can't see, you have disadvantage on the attack roll. This is true whether you're guessing the target's location or you're targeting a creature you can hear but not see. If the target isn't in the location you targeted, you automatically miss, but the DM typically just says that the attack missed, not whether you guessed the target's location correctly.
Will they have to ready actions to fight against it when it reveals itself on an attack?
While they do not have to do this, it is certainly one possible strategy. The Abishai will indeed reveal its location when it attacks, again through its sounds (ibid):
When a creature can't see you, you have advantage on attack rolls against it. If you are hidden--both unseen and unheard--when you make an attack, you give away your location when the attack hits or misses.
Be aware that readied actions will mean that the martial characters in your party will be at a further disadvantage, in that they will be able to make only one attack rather than their normal number of Extra Attacks.
Also, a readied action can be either a Move or an Attack, but not both. So unless the devil is revealed to be within their melee reach, their readied action will be wasted if they have a melee weapon in hand when the devil reveals itself by attacking. The devil is smart (+1 Int, +3 Wis) and can see perfectly well. Once the party starts using readied actions, it will certainly choose its targets to reveal itself while not in anyone's readied melee reach (if possible).
Another strategy, given the limited size of the darkness, is simply to use their movement until they 'hit' it (they are not permitted to enter a hostile creature's space), and then attack it once found. This is what my party usually does against invisible creatures indoors. You as a DM will need to rule on whether they can orient themselves or conduct a 'square by square' search in darkness.
In comments, @DarthPseudonym recommends that the characters Grapple the devil if they are able to discover its location. This has a number of advantages if it works - by reducing the devil's move to 0, they can hold it in a known location, or even drag it out of the darkness. The devil's best chance to break the grapple is Acrobatics with a Dex+3, and if the party has anyone with an Athletics skill they are likely to be able to beat that, especially since a martial character could get multiple Grapple attempts while the devil can only attempt escape once on its turn. While a great strategy once achieved, the difficult part is getting it to work, since they have to find the devil to begin with, and since readied-action Grapples are limited to a single attempt per action at things already within melee range, and the devil knows this.
If the devil fails to hide from even 1 of my players do all players know the location of it or only that player?
If a character (not a player) knows the location of the devil, that does not automatically communicate that knowledge to the other characters. However, the player could ask to do so. Then it is up to you as a DM to rule on whether such a communication requires an action or not, and how successful it is. "It's over here!" and even "Five (feet) to my left" are short enough to fall within six seconds, but are they understandable if the other party members can't see? Further, if players wish to communicate information gleaned from their Passive Perception success against the devil's Stealth, they might have to wait until their turn to do so.
Players inside the sphere are blinded, but what about players outside the sphere trying to fight something inside it?
See the quotes on attacking Unseen targets, above.
