"What exactly prevents a nominee from saying what he/she believes, whether it be about abortion rights, death penalty or the legality of trading baseball cards?"

A judge should believe in the law. It should not matter what a judge believes regarding "abortion rights". For example, if I believe that all drugs should not be illegal and I'm a judge, my job is the determine whether someone did break the law regarding drug use, not to change the law regarding drug use. It is the function of Congress and Executive branch to make laws.
There is some leeway in this principle, but it's always a slippery slope. An example is that if it's in the judgement of judge, that people really want a different law, but the legislative branch is incapable of performing it's duty, a judge might resort to legislating from the bench.
It also depends upon how a case is argued- on both sides. And of course the legistative branch can not make laws which violate the constitution and balance of power of the three branches of govt. In general the govt serves the people, the constitution curbs the power of Govt and balances the power of govt as a further safeguard, the Judge does represents the people interests [though not elected directly] and of course the Judge is a citzen who must hold his duty of a citizen above all other considerations.
Of course I haven't really answered your question. How about, the Judge judges cases. Explanations of why he/she judged a particular case in the past should be completely valid type question. Future cases will depend on numerous factors, and require speculation, and the Senate hearing is not a court. The amount detailed answer required could function as a sort of like court ruling without actually being in court.