To answer your first question - yes it can work, you can live in a social context without hierarchy without identifying as an anarchist, non-hierarchal communities and societies existed well before anarchism arose as an ideology.
As for the second - no, there can be no third party, not in form that you seem to suggest, which leads me to the larger issue here - your question doesn't engage with anarchism on it's own terms. You're bringing a lot of assumptions to the table that have their origins in caricatures of anarchism, not in anarchist ideas and conceptions of social roles and organisation/ethics/morality.
Here are some of the assumptions I'm taking about:
"For anarchy to work, it must assume that there will be a large faction, if not a majority, of people who oppose anarchy and want government (initially, at least)."
- From the outset you're speaking to an 'after the revolution' scenario, which just isn't relevant to a lot of anarchists. You're assuming that anarchists have a prescriptive vision for society that involves 'the revolution' overthrowing the social, economic and political establishment, and replacing it with their own structures, which are imposed on everyone. Not all (and I doubt even most) anarchists subscribe to any of that. Not only are there many different models for resistance/attack against the state and other forms of hierarchy (such as insurrectionary anarchism), but there are also many different perspectives on the idea of a prescriptive vision for society - from what one should look like to whether it's useful to have a prescriptive vision at all.
"To assume that anarchy would work when there are only anarchists is meaningless"
- Following on from the last point; it may be meaningless to you, but for anarchists who instead of trying to foment global revolution focus their energy on creating anarchic social conditions around themselves, specifically with friends who hold similar values, desires and politics (ie. anarchists), the project of creating functioning anarchist communities populated by anarchists is far from meaningless, it's ongoing lived experience.
"No one has any right to impose one's own will over another individual, or another group."
- Yes, insofar as no one has any right to or from anything. What you've appealed to here is the 'non-aggression principle', or 'the harm principle', and you've couched it in the language of natural rights. The assumption anarchism is predicated on both of these is in my experience very common, maybe because liberalism occupies such a place of influence in our political culture, and the harm principle and natural rights constitute the framework within which liberalism conceptualizes 'liberty'. Nonetheless, that assumption is deeply inaccurate - while liberalism has influenced anarchism (particularly very early anglophone anarchism and certain strands of individualist anarchism), the non-aggression principle and natural rights are far from being univerally accepted amongst anarchists, and many reject both completely. For a detailed argument against the non-aggression principle and natural rights I highly recommend Feral Faun's essay 'The Cops In Our Heads' (link below), it's quite thorough and very readible.
"But one must admit that it makes things very convenient and streamlines the conflicts that inevitably rise when groups don't see eye to eye with one another."
No, I strongly disagree. Take a moment to consider the 'objective' 'third party' that currently exists in western societies - the judiciary and legal system. When disputes arise between individuals or more abstract agents like businesses, going to court is usually the last (or first, in cases of 'corporate bullying') resort because of the enormous costs in time and money for both parties. Once disputes are taken up by the legal system, convenience and streamlined conflict resolution are thrown out the window. Conflict resolution is usually a grueling, tedious, frustrating and painstaking process, regardless of what from the process takes, and imposing a supposedly objective authority on the process, or appealing to a supposedly objective authority only fucks shit up further - it complicates everything and subordinates both parties to an arbitrary judgment that they have little reason to submit to if they don't come out on top, which is why we have police and prisons - to coerce them into submission. Part of the problem is that in order to claim objectivity, especially for an institutional third party that is persistant beyond any one dispute, the set of rules and consequences for breaking them has to be arbitrary, it has to be insensitive to the wider context, immediate situation and differing perspectives, to preserve the legitimacy of its own objectivity. To me it seems that any appeal to objectivity is inherently hierarchical as it places an external above both the appealer's own being and sovereignty* and the being and sovereignty of whoever is on the receiving end.
* (in theory, although what is appealed to as objective is often a direct reflection of the the appealer's subjective perspective)
The Cops In Our Heads: