What role should the laity play in selecting a priest for their parish?
In the Orthodox model, the congregation does not own or run the individual parish or church. The property is the property of the bishop. The clergy are the extension of the bishop. The bishop alone decides who he will ordain and where he will serve.
Actually I don't know that I agree about this being the case in practice. The church property actually does belong to the people. The land and the building belong to the people, the liturgical items do not (e.g. gospel book, chalice, atimens, etc) but the vast majority of the rest belongs to the people.
This is incorrect in practice, in theology, in canon law, and in US law. The parish itself as a corporate entity -- not "the people" -- holds title to the property and acts as a steward
(or an administrator) of the property. But its Parish Council, duly elected and
approved by the Metropolitan, may only exercise its stewardship when and so long as it keeps, practices, and proclaims the Orthodox Christian Faith pure and undefiled "in accordance with the holy canons, the Archdiocesan Charter and Regulations promulgated thereunder, and, as to canonical and ecclesiastical matters, by the decisions of the Holy Synod of the Ecumenical Patriarchate and the Eparchial Synod of the Archdiocese," (Article 15 of the UPR).
In all cases that I know of in which the Metropolitan has deemed that the Parish Council has failed to act accordingly, or tried to claim the property and building as its own, the Parish Council (which actually legally ceases to be the Parish Council in such cases) has lost in US courts of law. This is also true of similar cases in many different Christian churches (e.g. Episcopal, Roman Catholic, even Presbyterian).
So, in practice, the Metropolitan can and will dismiss errant Parish Councils and assume stewardship of the parish property under Article 16, as follows:
Sections 6: Based on the recommendation of the local Metropolitan, that a Parish is in heresy, schism, or defection from the Archdiocese, the Archbishop may declare the Parish in canonical disorder and may assume the administration of the Parish and control of its properties until the Archbishop, in consultation with the Metropolitan, declares the Parish to be in canonical order. In the event that the Archbishop, in consultation with the Metropolitan, determines that the Parish cannot be restored to canonical order, the title to properties shall vest in the Archdiocese. If under applicable law, title may not vest automatically in the Archdiocese, title shall vest in an ecclesiastical corporation controlled by the group of parishioners that the Archdiocese determines remains loyal to it.
Section 7: In the event that a Parish is deemed to be in canonical disorder and title to the Parish properties is vested in the Archdiocese, the Archbishop and the respective Hierarch are authorized to sign any deed or other document as may be necessary.
My question is, what do you do if a bishop assigns a priest to a parish and the parish refuses to pay the priest?
In all cases I know of such issues are agreed upon between the bishop and the parish council before the assignment is made official, so it shouldn't be an issue. However, hypothetically, if an agreement can't be reached, the bishop would need to deal with all parties to come to a resolution. If that can't happen, the bishop has the authority to dismiss the parish council if need be, although I doubt that would happen unless there were more to the story.