Honestly if i was allowed to play King maker of my little army to the extent i would like, i would found the Inquisition on a very simple concept: Ensuring stability to mainland Thedas at any and all cost necessary. The Creed of the Inquistion ultimately wouldn't be of race or faith but given those all have unifying principles it would be incorrect to claim that just because the Inquisition would welcome one and all, it wouldn't say deny entry for a select handful of groups. Imperium scum: Should be obvious why, Qunari, Again Obvious, Dalish elves, Basically any active or neutral hostile to the stability of Thedas.
The Inquisition under me wouldn't focus overly on any particular facet or faction overly, because ultimately it exists as an impartial peace keeping body, not only because threats posed to the world from the fade but also due to the stupidity, ill intent and greed of its mortal inhabitants.
To expand upon that concept let's say we are allowed to interact with the possible elven rebellion mentioned in Asunder assuming it is still ongoing within the game period. I wouldn't support the Elves with the Inquisition, in fact if the opportunity was offered i'd likely move to crush the rebellion as swiftly and brutally as possible, endearing myself one or both sides of the nobility of the conflict and urging them to commit at least a temporary peace to combat the true threat, aka the veil tears.
For any interaction with the Mage/Templar conflict the reasoning and methods are entirely different because the situation is different. While the Inquisition would try to not involve itself in that business if it became imperative to do so due to the chaos and strife it was causing Thedas, it would be imperative to move upon one or both factions involved. Both given the chaos of having unchained mages in such a world state, the Inquisition if forced into the conflict would likely side with the Templars. Once the military capacity of the mages to make war is reduced to cinders then the Inquisition can return to its place as a third party and attempt to assist in brokering a peace between the factions if required.
For Economics, the Inquisition if required would required would act as an intermediary between various Monarchies and Dwarven Trading houses and perhaps other factions, because strong trade is crucial for economic stability and economic stability is needed for national stability, National Stability is needed for Thedosian stability and as expressed earlier this Inquisition would exist to safeguard that balance. The Dwarves and Antivans likely would be the two largest backers of Inquisition aid due to their vast import in the majority of their respective trades, Antivans for ocean shipped goods and Dwarves for metal and stone works and Lyrium.
This Inquisition wouldn't aid any faction over another unless if it came into direct conflict with the group's mission statement.
Ultimately this Inquisition just like the prior one would exist to safeguard the world from it self and its threats.