Yes, I spelled the title correctly. These are common myths about the Latter Day Saint movement, a religious movement that contains The Church of Jesus Christ of Latter-day Saints (note the hyphenation; also commonly called the Mormons), the Community of Christ, the Church of Jesus Christ of Latter Day Saints (Strangite), the Church of Christ (Temple Lot), the Fundamentalist Church of Jesus Christ of Latter-day Saints, and many others. Of course, to be completely honest, this is mostly about the Church of Jesus Christ of Latter-day Saints (LDS) and the Community of Christ (CoC), formerly the Reorganized Church of Jesus Christ of Latter Day Saints and their common (and separate) history.
The Kirtland Temple exterior was constructed using the sacrifice of fine china and hair of the impoverished Sisters of the Church. There is evidence of glass and hair in the original stucco exterior, but it appears to be regular glass and horse hair.
The freezing of the Mississippi River in 1846 for the LDS exodus was miraculous. Not really – the river often freezes over this far north.
The Community of Christ membership is falling. This is both true and false. Membership in the United States is falling, but worldwide membership rates are doing well. Although it isn’t discussed much, the exact same situation applies to the LDS Church.
The Community of Christ gave women the Priesthood due to a lack of male leadership among their General Authorities / because they ran out of Smith descendants. Not true, but is related to the above myth. The priesthood was given by a revelation to the Church President. The current Church President is not a descendant of Joseph Smith, but the pattern was broken by revelation. The last Smith President is actually still alive, holding the office of Emeritus President of the Church. (There’s also a bizarre myth floating around that the reason for the change was that when the Church ran out of Smiths [untrue, remember] they looked at the genealogies and Spencer W. Kimball [or some other LDS prophet] was the actual Prophet by right of the RLDS Church. If you remember that none of Joseph Smith’s children ever joined with the LDS branch the absurdity of this story should be obvious. Our Church is full of Hyrum’s descendants.)
The LDS Church aided the CoC/RLDS Church at some recent point financially Always untrue. The relationship between our Churches hasn’t always been very pleasant (in some ways the RLDS Church began in opposition to the polygamy of the LDS Church); it’s only been in the last few generations that we’ve gotten along much. There apparently was some land that was swapped to help the CoC in the construction of their Temple in Independence, but the CoC has never received from or given financial assistance to the LDS Church. In fact, the CoC is rather financially secure and, after learning a bitter lesson about the dangers of debt in the construction of their Auditorium during the Depression, they never start any project unless they have all of the funds needed and never build on credit. The Independence Temple was already paid for before it was even built.
You can read many more myths about the relationship between the Community of Christ and the Church of Jesus Christ of Latter-day Saints here and here.