The Mormons believe it does not matter where or when you were baptized, that if you want to join their sect, you have to do it their way, even if you were a man or woman of God,baptised by a Baptist or Catholic priest, you have to have that special baptism of the Mormon temple.

Baptisms in Mormon temples are not for living people, but for people who have died who they believe live on in the spirit world (the place Jesus called "paradise" to the thief on the cross) who have accepted the gospel in the hereafter but were not able to receive the rites/ordinances of the same while on earth.

The rest is correct. Mormons believe that Christ did in fact establish a church anciently, and endowed that church with divine authority (called priesthood keys) with Peter being the presiding member of that church, endowed with those keys, just like the Catholics believe. But unlike the Catholics, and like some protestant sects who still acknowledge that Peter received such authority, they believe the Catholic church fell into apostasy and lost those keys of authority. They claim to know this because Joseph Smith said that God told him so, and that the resurrected Peter (among others) visited Joseph and bestowed those keys upon him.

So yes, Mormons do require you to get baptized again in their church in order to gain membership even if you had been baptized previously, as they do not believe whoever baptized you was properly authorized to do so. They see the ordinance of baptism, along with others that ARE performed in their temple, as necessary to enter the highest degree of heave in the afterlife (referred to as the "Celestial Kingdom"). But in a way they believe that all who are not considered "Sons of Perdition" (those who receive the Holy Ghost and a perfect knowledge of God and yet reject it) will be "saved" in the hereafter and be allowed into either in the "Telestial Kingdom" (lowest), and the "Terrestrial Kingdom" (next one up the ladder). But to enter the Celestial Kingdom, they do believe that certain ordinances are necessary, per God's word on the matter, and that said ordinances must be performed by someone who holds the authority to perform it, which keys they believe they hold exclusively in their church through Joseph Smith, through the resurrected Peter.

You can still be "saved" in Mormonism without receiving these ordinances, but Mormons have a very different idea of what "saved" means, and it includes the tiers I just described.

Baptisms for new members are not performed in the temples though, they're performed in a baptismal font within their normal meetinghouses--the buildings they all attend on Sundays in which non-members are invited to attend. To enter the Temple, they must pass a "worthiness interview" with their local bishop, however, who is a lot like a Catholic Priest, who Mormons go to confess their sins, and get a gameplan for helping them overcome such sins (whether they're forgiven or not is of course up to God, but their Bishops are their to provide spiritual guidance and support in helping them overcome said sins.

But yeah, to become a member of the Mormon/LDS Church, you need a Mormon/LDS baptism. But that baptism is performed in regular meetinghouses, not in their "temples." Those buildings are reserved for marriage ceremonies (called "sealings"), the receptions of something called an "endowment" (where they're given a new name and at which part they start wearing their "garments" under their clothes--what some like to call "magic underwear"), and baptisms and endowment ordinances for their ancestors. This is why Mormons are so obsessed with doing genealogy work. Each time they find someone in their ancestral line who has not received said ordinances in the temple, they go and do those ordinances for them on the off-chance that they have accepted the gospel in the "spirit world" (the place that Jesus referred to as "paradise" to the thief while on the cross).

Source: Me. I used to be Mormon, served a Mormon mission, and got married in a Mormon temple, so I know this stuff pretty well in depth.