1) How do the Ordo Malleus operate in the background?
Well, I'm pretty sure the current background would suffice e.g. look for daemons and kill it. Of course, I'm more of a militant-type of Inquisitor Lord and I'm sure someone (Masked Thespian) can explain better the more subtle approach of waging the war against Daemons.
I'm quite happy with the current background too. I don't really want to change too much either as I'm fairly happy with how the OM operate.
2) How should Daemonhunters play on the tabletop?
Like before, Inquisition should have a lot of techno-magic in their sleeves and some exotic Psychic Powers (mostly used against Daemons of course though I wouldn't mind 1 or 2 general-purpose Psychic Powers). They should have as many tricks as the Eldar whilst having an "Iron Fist" comparable to Space Marines (in the form of Grey Knights). Grey Knight would be Assault/Close-Range orientated; Storm Bolter and Nemesis Force Weapons as well as "Assault" Special/Heavy Weapons. Their Shrouding and "Teleport" Special Rules should be able to get them where they need to whilst protecting them from being getting killed by a Railgun 72" away. Summary, Eldar Marines 
I mostly agree. I've always seen the idea of Inquisitors as having a distinctively sneaky bent about them, especially the Radicals. Ergo, I would suggest that Inquisitors be similar in
function to Space Marine Librarians; warrior-Psykers capable of supporting other units but also taking charge and defeating the abhorrent (however, they shouldn't be capable of doing both at once, but have the option during army list building of doing either). Grey Knights, on the other hand, are the hard line needed when a certain something hits the fan. They are warrior-monks, capable of besting the worst the forces of Chaos can throw at them and should be able to overcome almost anything. I agree that Grey Knights should have a close range focus, able to do short range firefights as well as close combat.
3) Will there be any tangible gameplay difference between a Puritan and a Radical army?
Guess not since GW saw fit to tear away the IG Doctrine, Space Marine Traits and Eldar Craftworlds.
I disagree with this sentiment. Whilst Doctrines and Traits are gone, you can still see successors to those abilities. Marines get characters granting the whole army new abilities and the Guard get characters with unique sets of (and sometimes unique) Orders. Ergo, I feel there is enough room to distinguish between the two.
I'd already come up with this idea before Zenny did, but I would suggest we create a rule named
Factions; some units, wargear, weapons and psychic powers would have a label of
Puritan or
Radical and the
Factions rule would basically state that no Daemonhunters army may include both
Puritan and
Radical elements. There would be plenty of units and equipment WITHOUT either label allowing those things to be taken by both army types. Ergo, in a simple stroke, you have two distinct army themes; that of the militant, bombastic Puritans backed up by the forces of the Grey Knights, and those of the sneaky, underhanded Radicals whose power is based upon trickery, men and captured Daemonic power. Linking with the "Special Characters grant unique abilities" bit, you could even have some Special Characters granting one or two
specific pieces of the "opposite" Faction (representing Puritans in the process of turning Radical, for example). For example, you could have a Puritan Inquisitor Special Character armed with a Daemon Weapon, who is gradually turning him more Radical.
Heck, you could even have different versions of the same Special Character (in a similar manner to Manfred the Acolyte and Manfred the Count from Vampire Counts) representing older and more corrupt versions of the same man.
Examples of those with the
Puritan rule in the current Codex would probably include Grey Knights and Hierophants and those with the
Radical rule would be Daemonhosts and the Daemon Weapons detailed in that White Dwarf article.
I've not got long right now but I've got several other ideas that I will flesh out later but will mention now.
Daemonhosts: there could be two types of Daemonhosts; bound and unbound. Bound Daemonhosts would become Retinue members, granting Radical Inquisitors additional (and unique) Psychic powers and perhaps close combat ability. Unbound Daemonhosts would be Elites and could be Monstrous Creatures; amongst other things this gives Radicals some serious hitting power and makes playing them a distinctively difference experience to using Grey Knights. Obviously we wouldn't give them tabletop powers as described of background Daemonhosts (who have been known to destroy Warlord Titans with a glance) but make them representative of that power.
Grey Knights: These guys should be like the Tomb Kings of Psykers; relentless and unstoppable through a minor yet constant barrage of abilities. I would suggest allowing each unit to purchase a single psychic ability that didn't require a Psychic test. Each of these wouldn't be stupidly powerful, in a similar manner to an Eldar Warlock's power, but could represent things like The Shrouding (which I don't believe should be an army-wide special rule any more), Teleport training (granting the equivalent of Heroic Intervention) or something like that.
Inquisitors: These guys should have a huge range of things to choose from (like Lone Wolves and Wolf Guard get), but not be able to do everything at once. I think this is the key to Inquisitors; they need to be built with a specific function in mind. Each Faction would have unique equipment, psychic powers, weapons and retinue members, as well as a shared pool of all of those (for example, Hammerhand is a psychic power I'd imagine could be used by either faction but Banishment would likely be Puritan whilst Destroy Daemon be Radical).
