Right here goes my proposal.
1) How do the mages replace the financial subsidies of the Chantry
system? Right now, mages are fed, housed, clothed, and trained at
Church expense with pretty limited duties. Where does the money come
from without the Chantry?
The Formarri are currently the "merchants" of the Circle system. An expansion of that system around the nation the Circle is in could provide the needed currency. By selling potions to customers in need, providing healing services for a price and assuming that a Chantry free Circle won't abduct children who show magic powers, tutors could be paid for by nobles and wealthy residents who would prefer not having to send their child away. Of course necessary inspection of who would be the tutor would be undertaken to avoid letting a blood mage teach somebody like Jowan.
2) Who provides physical protection for the mages, which the templars are supposed to do?
I suppose some form of "New Templar" order would be formed under Circle supervision. Their job would be to guard mages, hunt apostates and deal with any demons and abominations. Tevinter could probably provide the means to training Templars as they do have them.
3) Do they have a Harrowing? Rites of Tranquility? Who deals with renegades and abominations if not the Templars?
There would still be a Harrowing to ensure the Mage can resist demons. The Rite would stil exist for trouble makers as a severe punishment, one that isn't given casually like the Kirkwall Circle. As for renegade and abominations the "New Templar Order' as I suggested above would deal with it along with perhaps Mages who have experience with fighting demons and can probably assist without being possessed or switching sides.
4)
What is their relationship to the political powers now that the Church
doesn't protect them from the lords and kings? Or are they the lords
and kings now? What about the mage fearing peasantry?
Their relation to political powers would be a servile one. Ferelden, Orlais, Antiva, etc would need Mages in the army to help fight other nations or darkspawn or Qunari. They would be expected to obey the law of the land and if not those in power have the right to punish the member of the Circle who broke the law or the Circle could handle judgement themselves.
Mage fearing peasantry would be tolerated. Attempts could be made to convince them of the good intentions of the Circle. If the peasants react violently perhaps the Circle could take an isolationist stance on villages that have a strong dislike of them.
5) How do they find the neophyte mages? How are they recruited for training?
Same way as before. The parents or someone reports them to the Circle. An emissary would be sent to meet with the parents and talk about the child's future. The child could either go to the Circle for training or if the parents can afford it the child could be tutored at home. If the parents are rather adamant about not wanting their child to go but can't afford it a deal could possibly be made. Such as a farmer could supply bread to the Circle to pay off the debt of having a mage tutor his or her child.
6)
Who governs the mages? If First Enchanter, what can the mages do if
that First Enchanter turns out to be Danarius instead of Irving? Who
can you appeal to without the Chantry overseeing the Templars
(theoretically)?
There would be a supreme council governing the Magi. If the First Enchanter is abusing their power then outside assistance or the Senior Enchanters may arrest the First Enchanter. To appeal the supreme council would be written to which would be made up of various Senior Enchanters from each Circle with the many fraternities having a voice in the council for fairness.
Thats my basic idea for it. If you have any questions or criticisms of it feel free to point them out and i'll do my best to answer.