It’s extraordinarily difficult to really say anything definitive about the Churches of Christ because they’re not all the same. The are very quick to tell a person that they are not a “denomination”,and recognize no authority beyond their own congregation.
Some have certain rules concerning music at their services. Some allow it. Some forbid it. Some have rules concerning the use of one or multiple cups for their communion. Some insist that only one cup be used for all, while some insist that individual cups be used. Individual congregations can be extremely idiosyncratic.
It can get confusing. I was once tangentially involved in a lawsuit between two congregations of the Church of Christ over a church building and piece of land. The original donor had specified that if ever the congregation quit doing those things (like the music and the cups…whichever side he took of it) the land should go to the next nearest congregation that believed as he did. Well, one congregation claimed the one with the land restrictions had departed from the true teachings, and that they should own the land and building.
The poor judge had to get a lot deeper into Church of Christ liturgical rules than I’m sure he ever imagined he would.
So, you really can’t say anything about the “Church of Christ” and have any real certainty that it applies to all.
Oh, at least the C of C people I know abhor Mormonism, and call it a “cult”. They weren’t started by joseph Smith, but by a near-contemporary of his whose name, I believe, was Campbell. There is some evidence that Joseph Smith admired and envied Campbell and patterned much of Mormonism on Campbell’s structure and teachings. So there is a common root there, but not much more.