Can a priest add his own prayers to the Mass after the dismissal? Can a priest add other public prayers to the Mass after the dismissal? If so, are those assisting at the Mass obliged to stay and pray those prayers with him?
I think the answer is partly in this communication by the Office for the Liturgical Celebrations of the Supreme Pontiff, titled "The Priest in the Concluding Rites of the Mass", issued in 2010.

First, let us recall the Rite in the Ordinary Form. As the text says, this is:


  "The concluding rites consist of 1) Brief announcements, if they are necessary; 2) The priest's greeting and blessing, which on certain days and occasions is enriched and expressed in the prayer over the People or another more solemn formula; 3) The dismissal of the people by the deacon or the priest, so that each may go out to do good works, praising and blessing God; 4) The kissing of the altar by the priest and the deacon, followed by a profound bow to the altar by the priest, the deacon, and the other ministers"


Thus, the final blessing could be expanded with the Prayer over the People (e.g see prayers for Lent here), or other "more solemn" formula.

The same text then goes on to present the goal of such Rite, where it is clear that this is a brief one. Some quotes:


  Of what has been said, it turns out that the two columns that sustain the Rites of Conclusion of the Mass are the blessing and the dismissal. ... From these biblical elements stems the Christian liturgical use of blessing, which has the meaning of "asking God for his gifts on his creatures, and of thanking him for the gifts already received". ... 


And:


  In the Rites of Conclusion of the Holy Mass the priest is still carrying out a priestly task, namely, of mediation between God and the faithful people. It is not only a question of greeting one another and agreeing to meet the next time, remembering the commitment perhaps during the week. The priest here invokes on the people the divine blessing, while in the name of the people he thanks God for the gifts already received by his kindness. Here also he acts "in persona Christi." Because of this, he does not say in the plural "may the omnipotent God bless us," or "the Mass is ended, let us go in peace." He speaks in the name of the Person of Christ and as minister of the Church, because of this he imparts the blessing, while invoking it, and he sends the faithful to the daily mission of life: "may God bless you" and "Go in peace." Through him, Christ and the Church charge the baptized with giving this daily witness of the Gospel. 


From the above it is clear that after communion, the final rites are to be brief, and still in a very solemn mood, with the Priest acting as a mediator. Surely the priest has the capacity to insert "another more solemn formula", but this does not mean everything. For example, to insert here a rather long set of prayers like the Rosary or to perform the Stations of the Cross during Lent would, in my view, conflict with the solemnity and purpose of the rite. In fact, in my experience these rather long prayers are usually celebrated before the mass. 

Now, when it comes to the Extraordinary Form of the mass, things might be slightly different. For example, it is common in this form to have, as part of the Concluding Rites, some rather long extra prayers. Blessing prayers are longer, and also the first verses of Gospel of John are read (see full prayers here). If the mass is a Low Mass, some final prayers to Mary and the Archangel Michael (known as Leonine prayers, and found here) can be added once the mass is finished. Whether more prayers can be added or not, I refer you to the 1962 missal (see in particular page 66, although text is in Latin).