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So, I've taken a crack at re-writing the inquisition index to fix some of the remaining flaws.

Small PDF File

New Units
There are now four new characters:
The Interrogator is a cheap HQ who can function as a low-level combat character or provide a little psychic support. They aren't great at anything, but can do a lot of jobs.
The Executioner is a slightly better version of the Ministorum Death Cult Assassin with a few options to slightly tweak how they perform in close combat.
The Gunslinger is the shooty equivalent to the Executioner; a character version of an acolyte with a focus on guns. Alternatively, a bargain Imperial Kellermorph.
The Sanctioned Psyker is a low-end psyker, equivalent to the Interrogator with all the combat stuff stripped away. They aren't as good as Primaris Psykers, but should be about on a par with Astropaths.
I've also split the current Acolytes into two separate units; Investigator Acolytes and Warrior Acolytes. The former are fairly cheap, give a deep-strike denial zone, and can have a few options which allow them to buff other units. They also have an anti-psyker option. Warrior Acolytes are obviously more about combat, with better stats and more options for weapons, as well being possible to upgrade with carapace armour to make proper Inquisitorial Stormtroopers.

General Gameplay Issues
I removed the existing 'Quarry' rule. I feel that it is bad rules design to make a free option arbitrarily good against some armies. Instead, each Ordo now has a Chapter-Tactics style rule, called Ordo Mandates. Each one gives a debuff effect to nearby enemy units.
The rules for combining Inquisition units with other Imperial ones have been tightened up a lot. You can still mix them in with thematically appropriate forces, but there are a lot more limitations. I'm fairly satisfied with these rules except that at the moment I have 'Astra Cartographica' down as one of the Ordo Xenos' potential allies. I feel that it is appropriate for Rogue Traders to work closely with the OX, but that faction currently only has two models in it. Likewise, the Officio Prefectus is basically just commissars and I think that the Militarum Tempestus might work well for one of the Ordos but I'm not sure which one (which pretty much says Ordo Minoris to me).
Daemonhosts and Jokaero are restricted to Malleus and Xenos respectively. You can still have one in a mixed detachment, but you lose your Ordo Mandate and won't be able to take other Imperial forces in the same detachment. The Grey Knights might be OK with an OM guy towing a daemonhost around, but the SoB are likely to object if a OH inquisitor shows up with one. Not sure how I feel about this solution; I may walk it back later.
I added a new 'quarry' rule (might need a different name to distinguish it from the old rule) which basically boosts non-psyker inquisitors and interrogators to make them actually worth taking. At the moment there is very little incentive not to make them all psykers. I'm not totally happy with this solution and later versions may either have two points values for those units or two completely separate datasheets for each one (a simple copy & paste and it isn't like there is a page count limit).

New Options
Terminator Inquisitors are not tied to Ordo Malleus and normal Inquisitors have more choices, including power armour, flak-equivalent armour if they prefer and either no force field or an improved one.
The weapon list has been expanded, mostly with low-end options like autopistols and shotguns. Digital weapons are now available for a few characters and a few xenos weapons are too.

Balance
Inquisitors are mostly as they were, since I don't think they were actually bad.
Acolytes are very slightly cheaper, but get about as expensive as they were when you give them boltguns or storm bolters. With the slightly improved stats of warrior acolytes, I feel this is OK.
Daemonhosts are identical to the index version.
Jokaero are very close to the index version. They got a melee weapon option, but they are still terrible at it so I didn't feel a price hike was justified.
Executioners are slightly more expensive than DCA, but they are characters and have more wounds. They are slightly cheaper than the star-striders character one. They lost the zealot rule, which may make them a little weaker, so maybe adding that back in or slightly dropping points might be necessary.
Gunslingers are much cheaper than kellermorphs, but also significantly weaker. Not sure what else to compare them to.
Sanctioned Psykers are priced similar to Astropaths, since I feel that is the closest equivalent.
Interrogators, I frankly just plucked a number out of the air. I guess you could see them as roughly equivalent to a sanctioned psyker plus a couple of warrior acolytes or one of the fighty characters.

Stuff I haven't done (yet)
Psychic Powers. I have some ideas, but for now the existing ones will do.
Stratagems. There are a lot of issues with the existing ones, but I felt it was better to get some feedback on what I had so far.
Unique Wargear. Again, not a high priority.
Background and illustrations. Honestly, I am unlikely to ever get around to these.

Looks like you beat me to rewriting the inquisition supplement, but you did a much better job than I would ever have done. I like how you are freeing up more weapon and equipment choices, I just wish you would go further, frankly.

To start with the equipment lists, I think it should be divided up into 5 simple lists, melee,force weapons, pistol (including the hotshot laspistol), ranged, grenade. Nothing should be restricted by ordo and it should be pretty obvious what goes were. There will be some restrictions in regards to terminator models, namely psycannons being termi only.

The main difference between units will then be what wargear they have access to, aside from special rules and stat line. Investigators may only take one weapon from anywhere in the list, Warriors (they need a better name) may take a pistol and a grenade in addition to the investigator's choices. Interrogators may take a grenade and two of anything in any combination (except two ranged weapons), and inquisitors may take two grenades and three of anything, limited by up to two melee weapons, up to two pistols, and up to 1 ranged weapon.

I, personally, would drop the gunslinger, executioner, and sanctioned psyker and instead allow the inquisition to take ecclesiarchal battle conclave units. I don't see the gunslinger and executioner adding anything that can't be replicated by existing units.

On to the particular wargear choices, are artificer bolt pistols 1 damage or 2? I wouldn't even include them and other mastercrafted weapons personally. I think of those as being for special characters and hyper-elite HQs, like SM captains.

While I'm on the list of things to remove, all of the special wargear should be added to the relics. The tome can remain as is, but the psyocculum can be more interesting. It should allow the inquisitor to shoot at an individual model, instead of unit. But if you chose to do so, roll 3d6. If there are any doubles, the bearer suffers the perils of the warp as if it were a psyker. If there are any triples, the bearer may shoot twice.

Now onto rules, I like the idea of ordo mandates, especially the ordo xenos and mallus one. I think the ordo hereticus mandate is infringing on the warload trait too much, it should just be based around judgment, and changed to something like every hit roll of 6 does an extra attack (no chaining attacks) in both shooting and melee. The ordo minoris, meanwhile, should probably have a chose-your-own-ability sheet like what the SM and eldar now have.

For unit rules, I like the investigtor's deny aura, I just think they should cost 6ppm. Warriors should go up to 7ppm too. Both acolytes should also go back to being elites and be able to be taken as characters. A mystic should be able to be substituted into in any acolyte unit. It will be a real psyker at 15 points (basically a substitute sanctioned psyker), knowing 1 additional power, but with the same 3d3 psychic test as the interrogator; which, by the way, should trigger perils on both a triple 3 and triple 1.

I don't really see many problems with the existing stratagems, aside from the orbital bombardment one, which is always in a bad spot.