To answer your questions point by point:
- Which primary mouse skill is most effective with Chakram for both generating hatred quickly and dealing damage in between Chakram shots?
All 4 of your primary skills generate the same amount of hatred. The only way to increase it so that you regenerate it faster is by finding a weapon that has a faster attack speed so you attack more often thus increasing your hatred generation. You can also find items that passively restore hatred as well, but I think that's outside of the scope of what you are asking for. So the shots that you are choosing will be dependent on what you are fighting. For example you'd want Hungering Arrow for single target high damage whereas you would use Bola shot for more Area of Effect in cases of clumped up groups of enemies.
- What action bar skills best complement Chakram?
This is a tricky one because it depends on your play style and if you are in a group or not. For example if you are in a group you would focus more on damaging than kiting as a solo player (although your companions could tank for you and you would still be damaging). So lets cover both.
*****Damaging***: Assuming you are at range and able to unload and it is a fight where Area of effect is needed (since you are using Chakrams). This also inherently assumes that you are in a group as well.
Skill 1: Shadow Power + Night Bane
This will restore health as well as generate hatred while it is active thus allowing you to fire off more Chakrams. Win/win. Caltrops and Smoke screen do also do damage, but require you to maneuver your character to have the Caltrops trigger, or damage cloud by the smoke screen. The maneuvering is one less action not spent firing a damaging ability.
Skill 2: Marked for Death + Grim Reaper(Area of Effect)/Mortal Enemy(Self Buff)/Death Toll (Single target)
Marked for death wins skill 2 hands down when it comes to damage output. The runes are your falvour or usage depending on the fight. If you again are fighting a lot of monsters, Grim Reaper will simply do more damage to more targets over time. Mortal Enemy gives you more hatred which again gives you more Chakrams to fire, and Death Toll is awesome if your group needs healing because this buff affects EVERYONE.
Skill 3: Sentry + Vigilant Watcher (Stationary fights)
For a pure damage perspective, you want Vigilant watcher as it extends the life of the turret, and assuming that the turret is in range of your target it will do significantly more damage because of the extra duration, which then in turn allows you to spend the discipline on other abilities. You could also take Aid station or Guardian turret as the rune, but that is dependent on the fight.
Evasive fire, fan of knives, also require you to maneuver in and around the target, see previous point about why this is bad from a pure damage perspective.
Spike trap is a viable alternative because the ability triggers where your mouse pointer is so this is a possible option depending on if you are dealing with single target or area of effect fights.
*****Soloing*** I'm going to assume the kite method for soloing as you will not be able to stand up and fight any large groups of mobs by yourself without dying. The idea behind this method is to be constantly moving while still firing (think stutter step micro with marines/marauders/medivacs if you play Starcraft 2). Side note: this is the style that I'm employing. What you'll be doing most of the time is clicking to move away, then pressing Shift and clicking. This forces you to stop and attack where you are to where your mouse is pointing. This is where the auto-targeting of Hungering Arrow comes in handy.
Skill 1: Caltrops + Hooked Spines/Jagged Spikes
The rune on this one is weapon/damage output dependent. If you are doing plenty of damage then Hooked spines will be better because you're interested in slowing them down, giving you maximum time to kill them with your weapon. However if your output is lower, the Jagged Spikes will help you kill them faster.
Skill 2: Vault + Trail of Cinders
Vault is used to get out of those sticky situations or to quickly form a backup line after dropping your Caltrops. The AI generally will follow you in a straight line so the trail of cinders, combined with your Caltrops means they will be moving slowly towards you, all the while taking damage (even more if you took Jagged Spikes).
Skill 3: I feel this one is a toss up between Spike Trap and Sentry
The reasons for Sentry were covered above so I'll cover Spike trap here. Because you are kiting, you can easily manipulate the groups of enemies so they move over your traps so they take damage from them. The same can also be said about keeping them around your sentry. But if we looking at the spike trap, either Long Fuse or Lightning Rod would be the runes that I would use for higher damage. Also, this trap would most likely be placed on top of Caltrops.
Skill 4: This is a tough one because they all do damage. I think this falls into play style again and fight dependent. Also, all these abilities require hatred rather than discipline. This one applies to both Soloing and Damaging in a group.
Strafe + Stinging steel for single target and Demolition for Area of Effect.
Strafe allows you to fire on the run, awesome for fights that require a lot of mobility. And again the rune depends on if the fight requires more Area of Effect or single target.
Multishot + Fire at Will(More damage)/Full Broadside(More damage)/Arsenal(Area of Effect) (I would only take this one as a secondary Area of Effect ability)
Fire at will allows you to do more damage because of the reduced hatred cost as opposed to Arsenal while does more damage in a localized location. Suppression fire may also be useful as it restores discipline allowing you to use other abilities.
Cluster Arrow + Any rune
This ability is more Area of Effect damage, I would only take it if there were a lot of monsters that need killing. This one also has a rune that has a heal component so it can be very helpful in a fight where healing is needed.
Rain of Vengeance + Any rune
This may be the best one to use as it doesn't cost any hatred and works on a 30 second cooldown. All the runes increase damage done, so again it is fight dependent. After some more personal play time rain is my favourite to use, mostly for the fact that it doesn't cost any hatred and is simply a cooldown. It may not be the highest damage, but it certainly allows for more usage of other abilities.
- Are there any passive skills that will can improve the use of Chakram by either increasing damage or attack speed?
Archery pretty much needs to be taken regardless of what your other two are.
Damaging
Custom Engineering would also be a good compliment to the damage section of this build as it requires less of your discipline so you can keep using it on Shadow power.
Steady Aim is also a good skill as it compliments the firing from long range method.
Night stalker, Sharpshooter, and Archery (with hand crossbows) provide really good synergy that will allow you to use your Action skills more often. You gain discipline when you crit, archery and sharpshooter increases the chances that you can crit.
You could replace Night stalker with Steady Aim if you have an excess of Discipline.
Soloing
Cull of the weak would be great because your targets will be slowed by your Caltrops so you do increased damage, would not be useful in a group.
Custom Engineering is again a good one because it increases the duration of your Caltrops and traps.
Best Answer
Many of these types of effects in games are learned from simply playing the game more. I don't envision these being available from an official Blizzard source, since it would be quite overwhelming for most to make use of the information - save the most advanced players.
The other side effect of this is that it encourages players to explore more skills and mechanics to master the game.