So I'm mildly excited, because this is the first Khawarij thread I have seen on the site since I joined.
Now one thing that would be effective is to outline who the Khawarij (of the first Fitnah) were and what did they believe in matters of Fiqh and Aqeedah. First of all, most of the men/women who were among this group considered the term Khawarij to be derogatory and preferred the term (as modern Ibadhi call them) Shurha, meaning the buyers. Which in context means the ones who sold the Dunya (earth) for Firdaus (Paradise) and has traded one for the other. So for the rest of my response I will attempt to refer to them as the Shurha.
Now the first tenant of the Shurha was that law and arbitration is for Allah alone, anyone who rules or decides something on his or her own intellect or reason is commiting Kufr Akbar and that person is a Kufr through and through. It's difference with Sunni or Shia versions is the level in which one is a Kufr and how quickly one can be called such. For instance, Shurha did not use the term 'Murtad' for an apostate or one who says the Shahada but whose actions is not of Islam, but instead escalated regardless of the persons Aqeedah to open takfir on that person. For their defense the Shurha pulled this surah as its war call:
Quran 6:57
Say "Indeed, I am on clear evidence from my Lord, and you have denied it. I do not have that which you are impatient. The decision is only for Allah. He relates the truth, and He is the best of the deciders."
This would entail that in a Shurha state or Caliphate, that rule by the prince would be that in which the prince is among the people, in their own words. Thus, ruling would be in many ways closer to a democracy than any other Islamic state until modern times, one in which the ruler is held accountable (more so) to the Shariah and decadence removed at all times, this in fact can be seen in modern times in Oman and the rise of Sultan Qaboos.
With the rule of Allah made as the base of the belief, the Shurha expanded upon this with what Salafi call Biddah. Examples of these are:
1. The Shurha proclaimed that takfir was permissible if one does not follow the Shariah in all manners without clear cut evidence from the entirety of sources, and removed the need for an Ulema to proclaim the Takfir.
2. That the piety of a man decided if he was the Caliph rather than the Caliph having to be among the Ahl ul-Bait or among the Quryaish, this is characterized in how the call of the Shurha during the Zanj rebellion was that an African slave is equal to any Arab if his piety surpasses the Arabs. As well, the decision making process is without the Ulema but decided amongst themselves based on piety.
3. The lack of a need for an Ulema. This is again part of the Shurha and their egalitarianism. It is noted that the Shurha proclaimed that the Ulema and it's entirety was Kufr and that they had disobeyed the Shariah. This lack of an Ulema creates in a way a more free response to religious questions within a society, without being under the control of fatwas from Ulema who might or might not pass it with the greater good involved.
4. The Shurha claimed that rebellion against the Wilayet or the Caliph is permissible in all cases without debate or the fatwa of the Ulema. Basically no wait, just rebel when you see one who does other than what Allah hath willed.
5. Then the practice of Kitman, which is a hallmark of the Shurha movement. Kitman means concealment and is different than that of the Shia Taqqiyyah, but Kitman is more like say camouflage in order to strike out in a short time, whereas as Taqqiyyah is concealment fir the sake of defense of ones faith. Kitman was what facilitated the Shurha revolts, getting close to Wilayet in an area while inciting the populace and in opposition to the Wilayet that you are paying service to.
Therefore a Shurha caliphate after the first Fitnah would practice a significantly modified caliphate from that of the Umayyads and would practice in a way a more effective method of assimilation of conquered people to any other Islamic state, with near constant jihad this state would have enormous growth potential. However the state would most likely prove to be enormously unstable and would likely lead to Wilayet warring amongst each other constantly over the title of the Amir Al-Mu'minin, just as Ali did with Muawiyah. In ways, the Shurha believed the war of the Fitnah was just, because as they say, what better way to decide Allah's wishes than on the field of battle in Jihad against ones enemy?
Now idk what the question exactly was from the poster but this is what I felt like typing lol.