IMO, the balance should work something like this:
Slingers are best as anti-skirmishers and cavalry snipers. They should be able to take out lightly armored troops quickly, and also are good at sniping cavalry from afar. Slingers should easily defeat javelin-skirmishers, and should also beat archers in a straight 1v1 fight. However, IMO slingers should have low AP damage & shouldn't do much damage against troops with heavy armor. An exception to this might be slingers with lead bullets and/or heavy shot. Units being shot by slingers should have significant morale penalties, but less of a penalty than if under fire by archers. Slingers should get an AP damage bonus if they are very close to an opponent to simulate how much more lethal slings are up close. Silven's mod used point blank range bonuses to simulate this if you'd like an example.
Javelins are best used against infantry, period. They are not very good at other roles, but their high AP damage relative to other missile units makes them the only ranged unit that is really going to be able to kill heavy infantry quickly. They are incredibly deadly if they are able to release javelins into an enemy's flank. IMO, a javelin to the back should be insta-kill...that ish would hurt & they are designed to penetrate armor and shields.
Archers are best used as a support role against the enemy body of infantry. While they won't be able to kill as many soldiers outright like javelin-men, they will cause a higher penalty to morale, have much longer range, and have much more ammo than javelins. Archers are also very effective against cavalry, if they are so silly as to sit around in range. Archers can hold their own for a short while against slingers, but ultimately won't be able to hold up 1v1 against the more pinpoint accuracy of the slingers. However, the one time that archers will be able to beat slingers is when they are able to make use of their firing arc. As slingers fire in a relatively shallow arc, they can't fire well over lines of troops or walls. Loose formation is especially effective against archers. Archers should be able to easily destroy javelin-skirmishers. I think archers should have very low AP damage, maybe only composite bows and longbows should have any AP damage at all. However, with the reduction in damage, you would need to make sure that the morale and cohesion penalties for a unit under fire ensure that archers have a role to play. Archers should also get a slight AP damage bonus at point blank range (to simulate higher energy of an arrow firing straight on rather than at an arc), but not as much of a bonus as slingers.
Slingers beat archers & slingers beat javelins. However, javelins and archers are both more effective against enemy infantry & cavalry than slingers - javelins at killing and archers at suppression. Archers and javelins can both fire at a higher angle of attack than slingers, making them more effective behind the lines. Slingers pretty much need to be out front or along the flanks to have good effectiveness.
Just some thoughts
