And by what authority does any church question people who state their beliefs in public?
And to answer your first question: yes, I place politics above church authority in matters of coercion, force, intimidation, etc.
If a church wishes to set standards for membership and question members about their qualifications, that's not a problem. But if a person wishes not to be a member, or wishes to publicly question the beliefs of a church, the church has no business questioning anything except that person's membership.
The essential purpose of the State is to protect the welfare of its citizens. If you consider the salvation of each inhabitant through belief in the true religion revealed by God part of that "welfare," then the State has an obligation to support the true religion and repress public dissidence against it while allowing those who wish to privately hold their erroneous views to do so. This is the way that Christendom operated until the advent of Protestantism, which denied the authority of the sole possessor and guardian of the true religion, the Catholic Church, created a rift between the two arms of society and eventually found enshrinement in the Freemasonic notion of "separation of Church and State." This, as you so aptly demonstrate, has led to the subjugation of the Church to the State, essentially turning true authority upside down and placing man before God. We are paying for it in spades.
It is my understanding that the people tried by the Inquisition were not just "questioning the Church's beliefs", but actively teaching something different from what the Church was teaching. This led to confusion, and the Church was concerned, as it is now, about regular Catholics being confused and living in a way that is contrary to the Faith, but not realizing it because of what they've been taught by their Pastor, local Church leaders, etc. Then, as now, the Church was concerned about the state of their souls.
This is the reason for the brouhaha about pro-abortion politicians receiving Communion. It is a Sacrament that is a public statement of one's 'communion' with the teachings of the Church. It causes scandal and confusion among Catholics if they see such high profile people being welcomed at the altar for Holy Communion. They may begin to think that being pro-abortion must be OK after all if these folks are not being rebuked by the Church. And, make no mistake, these folks were not rebuked in public before having been done so in private. All this started because some politicians decided to make public the private communication of their Bishops. After that, the Bishops had the duty to comment publicly as well.