Does the Constitution give the right of tax exemption to the Church?

The Antiestablishment Clause in the First Amendment means that no right can be given to any one church. The right is given to all organizations, religious or quasi-religious, providing that they meet certain requirements. For example, there are limits on income from business activities. If this were not true, you could avoid corporate taxes by affiliating with a church. By separation of church and state, it follows that organizations that serve political functions should not be treated as religions.

Not a constitutional issue, but a congressional issue.

Churches declare themselves to be non-profit entities. The tax laws do not require taxation of non-profit entities in certain categories, and the churches happen to fall into one such category. The constitution gives Congress the power to levy taxes and fees as needed (it's worded a bit more precisely than that, I'm speaking very generally here). It was and still is up to Congress as to how to actually implement that taxation.

You need to understand the difference between a constitutional permission and an "enabling act" to actually implement the thing that is permitted. For instance, the constitution provides for various kinds of equality under the law, but it took a (Congressional) Civil Rights Act of 1964 to enumerate and elaborate on the detailed laws encouraged by the constitutional amendment.

In your additional details, you make a terribly bad assumption. The state is a secular power. The church is not a single denomination so there is no way for government to choose a religion. There is NOT a CHURCH. There are CHURCHES plural. None of which is monolithic enough to warrant that level of treatment. The church is NOT equal to the state. The government clearly and always outranks the churches.(Remember "Render unto Caesar that which is Caesar's" ??? And "My kingdom is not of this world?" Well, government IS of this world and that is where you happen to live, bubbulah!)

The government tells the churches that to maintain their status as a non-profit, tax-exempt organization, they must not take actions that make them look like a different class of organization such as a political action committee. The latter is also non-profit but has a different tax status.

That is a right the people gave to the government. That's where the government gets off. And the people outrank both the government AND the churches (PLURAL). Let's put it another way. If the churches want to meddle in affairs of government, they INVITE government to meddle in the affairs of churches. Do you really want that?

Really, think about something before you open your mouth and reveal your lack of practical understanding. You'll embarass yourself less often if you just think first.

You misunderstand how it works.

All non-profit organizations are tax-exempt, not just churches. Taxes are collected because a business or person who makes a profit as a result of existing in this country must pay for the benefits afforded to him, her or it. Entities that pay taxes have a say in the government which they are supporting with their tax dollars. Churches get no voting rights, they can't participate in how the govt. is run, how laws are made, nothing. This is the right and privilege of businesses and citizens that profit and benefit directly from the govt. they support.

It can always be argued that a church is making a profit, and if provable, they can lose their tax-exempt status, in which case they would have to pay taxes, but thanks to the current interpretation of the Constitution, they still couldn't influence legislation.

The rights that churches do get have more to do with how the society values them, since churches generally serve an important social function for most people. Citizens will basically petition government to make sure churches survive and prosper. This is why churches get certain trivial privileges, like traffic law and parking suspensions on Sundays, etc.

Churches and other non-profits basically have to play the hand they are dealt, so that is why they are exempt.

Tax exempt status is a privilege, not a right. In exchange for it, a church or other group agrees to certain things. In a Church or organization cannot truthfully make this agreement, it has no business claiming exemption.

Not all churches do make that claim.

501(c)3 status does not make a group a church. It makes a group a non-profit organization.

Yes, they are a profitable organization that receives legal tender; whether it is or isn't threw charity does not matter.