The Father, Son, and Holy Ghost are three distinct beings.
In Latter-day Saint theology, Heavenly Father, Jesus Christ, and the Holy Ghost are three distinct individuals. Doctrine and Covenants 130, a revelation received by Joseph Smith, states the following:
The Father has a body of flesh and bones as tangible as man’s; the Son also; but the Holy Ghost has not a body of flesh and bones, but is a personage of Spirit. (D&C 130:22)
Heavenly Father had a glorified physical body from the before the world was created. Jesus Christ, however, did not have a physical body until his mortal ministry (Ether 3:16). Now, as a resurrected being, He has a glorified physical body like Heavenly Father's.
We don't know a lot about Heavenly Mother (pretty much everything we know is in the article you linked above), but from what we do know, we can presume that her physical nature is similar to Heavenly Father's – already with a glorified physical body from before this world was created. We know from the scriptures that once we receive a resurrected body, the spirit and body will never again be divided (D&C 138:17). Heavenly Mother could not have been born physically again as Mary (we can only have one body at a time!). Mary, despite her honored calling, was a spirit child of heavenly parents like the rest of us.
Latter-day Saints pray to Heavenly Father, in the name of Jesus Christ.
Latter-day Saints worship both Heavenly Father and Jesus Christ (especially in hymns, where either may be addressed). But when we pray, we only pray to Heavenly Father, and we do it in the name of Jesus Christ. From an article on LDS.org:
The Church's first article of faith states, "We believe in God, the Eternal Father, and in His Son, Jesus Christ, and in the Holy Ghost." These three beings make up the Godhead. They preside over this world and all other creations of our Father in Heaven.
The Mormon view of the members of the Godhead corresponds in a number of ways with the views of others in the Christian world, but with significant differences. Latter-day Saints pray to God the Father in the name of Jesus Christ. They acknowledge the Father as the ultimate object of their worship, the Son as Lord and Redeemer, and the Holy Spirit as the messenger and revealer of the Father and the Son. But where Latter-day Saints differ from other Christian religions is in their belief that God and Jesus Christ are glorified, physical beings and that each member of the Godhead is a separate being. ("Godhead," LDS.org)
Elohim is a name or title that refers to Heavenly Father.
Usually the name Elohim is used to refer to Heavenly Father, and Jehovah refers to Jesus Christ (especially the pre-mortal Jesus Christ). From the Encyclopedia of Mormonism (not an official Church publication, but generally a reliable resource):
Elohim appears in the Hebrew Bible as a common noun identifying Israel's God. ... It was also frequently used interchangeably with Jehovah, the proper name for Israel's God. ... Latter-day Saints use the name Elohim in a more restrictive sense as a proper name-title identifying the Father in Heaven. ("Elohim," Encyclopedia of Mormonism)