What are the rules of establishing public Treasury in the Islamic Law? What is meant by public Treasury? What are the common rules for its establishment?

Basically bayt al-maal (Public Treasury) in the Islamic state is the body where public money - of charities, Al-Fai` (Land and toll taxes paid by non-Muslims to the Islamic state according to an agreement), booty and other rights for Allah in money are kept, in order to spend it legally. It is permissible for the Islamic community to set up something like that even if it does not have the state power or authority as long as the applied systems in the society allow it. Consequently, it becomes a non-for-profit organization for collecting such money through which charitable projects supporting public interests are subsidized.

The channels of spending  zakaah are specified in the Book of Allah the Almighty in which Allah says, {Alms are for the poor and the needy, and those employed to administer the (funds); for those whose hearts have been (recently) reconciled (to the truth); for those in bondage and in debt; in the cause of Allah; and for the wayfarer: (thus is it) ordained by Allah, and Allah is full of knowledge and wisdom}.

Zakah supplies must be allocated for spending  in theses channels. Moreover, other supplies should be directed to the public interests without allocating any of it to the eight channels - as such a matter is wider than that of zakah. Besides, the issue of Public property is extremely sensitive in shari`ah as the Prophet (P.B.U.H) said, {those who devour the money of Allah with no right, the Fire of the Hell will be for them on the Day of Judgment}.
The difference between caliphate and ruling is that: in caliphate money is gathered and spent lawfully. However, if any violation occurs, caliphate will turn into ruling. And Allah the Almighty knows best.