Q:

How was the Roman Princeps Senatus appointed?

In Republican Rome, the princeps senatus (~head of the senate) was supposed to have been nominated by the censors, who were elected every 5 years for 1 and a half years.
What happened when the censors disagreed on the candidates?
Did one censor get to choose first (patrician censor, censor elected with more votes...), or did the senate make a choice and ask the censors to follow?
What happened if a princeps senatus died while there were no censors? Did the consuls choose the new princeps senatus, did the senate do it, or did they have to wait until new censors are elected?

A:

It is not necessarily be problematic if the censors disagreed. Scholars have generally thought that only one censor was chosen by lot to nominate the Princeps Senatus alone. If correct, then in the event of disagreements between the censors, the chosen one would have the final say. Much support for this theory is inferred from the 209 dispute (see below), where one censor was reported to have the choice "by lot".

The role of appointing the princeps senatus was given by lot to one of the censors. The other censor might act in harmony, but legally the one who was given the lot was completely free to choose the man he preferred.
 - Ancient Society. Vol. 5-6. Katholieke Universiteit Te Leuven, 1974. 210.

Granted, this is not undisputed, since most records (where there was any indication at all) seem to suggest both censors agreed on the candidate. One explanation is that the choice belong to one censor, but he is obliged to seek his colleague's consent. Alternatively, it has been theorised that normally, the censors make the appointment jointly, but if the censors could not agree, then the dispute is resolved by lot.

Previous scholars agree that the princeps was apoointed by one censor ...  [But] it cannot be coincidence that the censorship of 209 provides our only example of a dispute de principle legendo and our only instance of sortition in the choice of a princeps. The evidence for the conduct of comitia suggests that there was a resort to the lot (sortitio) only when an agreement (comparatio) could not be made.
 - Ryan, Francis X. Rank and Participation in the Republican Senate. Franz Steiner Verlag, 1998.

Unfortunately we can't know for certain how the nomination operated. Nonetheless, the prevailing understanding of scholars is that a dispute between the censors would be satisfactorily resolved through allotment.

If there was no censors when the position of princeps became vacant, then it appears that no replacement was appointed until the censorship was filled again.

As there was no censorship between 86 and 70, it follows that the post of princeps senatus remained vacant from c. 79/78 at least until the censorship of 70.
 - Tansey, P. "The Princeps Senatus in the Last Decades of the Republic." Chiron 30 (2000): 15-30.

Prior to 209, the oldest living former censor was by tradition appointed the princeps senatus. As a matter of custom, this patrician would be re-nominated to the position until his death, at which point the next most senior former censor succeeds him.
In this year, the first recorded instance of the censors disagreeing on who to nominate as the princeps occurred. M. Cornelius Cethegus argued in support of the tradition of appointing the senior ex-censor, whereas his colleague P. Sempronius Tuditanus supported Q. Fabius Maximus, who was censor in 230. T. Manlius Torquatus, the traditionalist candidate, was senior to Fabius by one year.
After an apparently fierce debate, Sempronius prevailed and Fabius became princeps senatus. The episode is reported by Livy in his Ab Urbe Condita Libri:

The revision of the list of the senate was delayed by a dispute between the censors in regard to the choice of a princeps senatus. The choice belonged to Sempronius; but Cornelius said that they must follow the traditional custom of the senate, namely, to choose as princeps the man who, among the living, had been censor first. That was Titus Manlius Torquatus. Sempronius claimed that if the gods had given a man the choice by lot, they also gave him an unrestricted right; he would make the choice according to his own judgment, and would choose Quintus Fabius Maximus, whom he could prove, even with Hannibal as judge, to be at that time the first citizen of the Roman state. After the war of words had lasted long, his colleague was giving way, and Sempronius chose Quintus Fabius Maximus, the consul, as princeps senatus (Livy, 27.11 9-12).

