No. The only for-profit ventures of the Church of Jesus Christ of Latter-day Saints (commonly called the "Mormon" church) are:
Deseret Book, a religious book and gift store
Temple Square Hospitality, a collection of cafe's and reception centers in Salt Lake City
Deseret Management, which rents out apartments and office space in Salt Lake City
Bonneville International, a newsmedia outlet
Beneficial Life Insurance (which is no longer accepting new policy holders)
There is no connection between the Church of Jesus Christ of Latter-day Saints (the "Mormon" church) and Melaleuca, aside from the fact that the current owner of Melaleuca, Frank Vandersloot, joined the Church at age 17. The Church itself has nothing to do with the company whatsoever.
Yes. Every genealogy website except familysearch.org is non-Mormon. Familysearch is owned by the Mormon church.
Melaleuca hnatiukii is an Australian shrub.
No it isn't.
No
Melaleuca viminea was created in 1839.
Melaleuca minutifolia was created in 1859.
Melaleuca elliptica was created in 1806.
Melaleuca penicula was created in 1990.
no
The word Deseret is found in the Book of Mormon and supposedly means Honeybee. Early Mormons adopted this word and the symbol of the beehive to represent their industriousness - they were busy, hardworking, and highly organized, just like bees.The Deseret title can still be found on many things related to or owned by the church such as Deseret News (a Mormon-owned newspaper), Deseret Industries (a Mormon owned thrift store), and Deseret Book (a Mormon book store).
In order to start with Melaleuca you would need have someone invite you or you can contact the Melaleuca 1800 number. See the related link for further information.