I don't think lack of education will be a problem. Even if you are the poorest mage, the College of Enchanters run by Fiona won't just let them go around without a basic understanding of magic for fear of abominations erupting. Even Tevinter teaches their slaves (e.g. Calpernia) the basics of magic out of principle. If they can't afford to live in ivory towers like the Circles of old, then the very least the College can set up apprentice programs under hedge mages like the Avvar Augurs or Chasind Witches. All that is required is the transportation and really, with all that magic, the College should be able to whip something up.
As for funds, these are people who can enchant items, reshape reality, and create magical wonders. Are you telling me the College of Magi can't market that without the mundanes to hold their hands? Have you ever read http://Codex entry: Darktown's Deal? The Dwarven's Merchant's Guild was able to make a business relationship with Orzammar who as surfacer dwarves are anethema to dwarven nobles. Yet they made it work. Why can't they just do the same thing? Have enchanting guilds, mercenary guilds, alchemist guilds sell their services to a discrete middleman (Mages' Collective) and all parties benefit. Now muggles could theoretically stubbornly war against this, but is it really a good idea to alienate Thedas' greatest scientists, weapons of war, and scholars? Those kind of jackasses would surely have enemies that the mages would rather sell to. In fact, for the very reason you just described, it gives incentive to mage solidarity in these sorts of things. You have the Aquetarians who would cross arms with Libertarians just for the uniformal adequate education of mages and libertarians will want to show the muggles that the system works to keep their freedom and the lucrosians can earn so much more gold as an independent marketer of their services than they ever could under the Chantry or templars' rigorous regulations. Other than the loyalists, I don't see how this doesn't benefit everyone.
Nah. When bioware has a game dedicated to an organization, that organization is going to be a big mover and shaker for the rest of the franchise, especially when it's been brought back from the grave and becomes a zero to hero story. In fact, as the Templar Order is disbanded, if the mages who hire their own templars (probably something like sellswords) don't handle abominations and demons, the Inquisition could take the role of like the Seekers of Truth to the College. Watching for corruption.
First, on the matter of universal mage education:
1) Tevinter educates all mages because the mage education there is run by the state, which has a great number of places from which it can get money, like taxes and trade. The state has stable enough funding to educate all mage children.
2) The College is an independent organization. It must rely on money from contracts and the production of goods, some of which will be in competition with goods from existing sources and others would be contested when more mage schools formed. The College would eventually not have a stable enough source of income to make mage education universal.
3) I don't hold a great deal of confidence in Fiona's ability as a leader and administrator and leader, but even ignoring that, I said that the College would become corrupt within a generation or two. It doesn't need to be Fiona. It is extremely improbable that the College will be graced with an endless string of selfless and thoughtful rulers, and when dealing with the education of all those mages in southern Thedas, it wouldn't take many poor decisions to throw everything into chaos.
Funding was also primarily an issue in my model due to competition between mage schools. They'd be competing for the same customer base and would seek alternate sources of income to make up for the money lost to other mage schools. And the mages would certainly face opposition from mundanes. Yes, they could look to the opposition's enemies to fund them instead, but once again, there would eventually be many schools competing for this same pool of clients.
And....um.... yes. One of the three scenarios that I listed will occur. The Inquisition is basically in control of all of southern Thedas. Such an organization with such a variable set of tenants and an entirely variable leader cannot be successfully translated across multiple installments. The best case scenario to back up what you are saying is that the Inquisitor will get the boot, of even more infuriatingly, disappear. Upon the Inquisitor's disappearance, it's pretty much up in the air what'll happen to the Inquisition.
Look, I know that you don't want that to happen. No one wants that to happen. But it's going to happen. It has to happen. Bioware even worked to cover themselves in game, with all those talks about how the Inquisition would have to disband.