What do Latter Day Saints believe happens to persons who have rejected Christ Jesus The Church of Jesus Christ of Latter Day Saints says this in their third article of faith:


We believe that through the Atonement of Christ, all mankind may be saved, by obedience to the laws and ordinances of the Gospel.  http://www.mormon.org.uk/beliefs/articles-of-faith


I also found this statement while I was doing research into the subject:

As spirits, we will have an interim judgment which will determine whether our spirits go to Paradise (where the Savior went when his body was in the tomb) or to the Spirit Prison called Hell.

Is this Spirit Prison called Hell a place of eternal torment, or do those who have rejected Christ Jesus get a second chance to be saved?
On the subject of unrepentant sinners and the Telestial Kingdom I found this quote:

The Telestial Kingdom will be populated with the unrepentant sinners who, in spite of the fact that they did ultimately acknowledge Jesus to be the Christ, refused to accept His teachings or His atoning sacrifice on their behalf. They were too proud or too rebellious to care to commit themselves to Him in any way, and were willing to accept the consequences for their refusal. They will spend eternity as “servants of the Most High but where God and Christ dwell they cannot come…” Doctrine &amp; Covenants 76:112.

I would also be interested to find out if the beliefs of Latter Day Saints on this subject have any basis in the words of Jesus as recorded in the New Testament.
The Church of Jesus Christ of Latter-day Saints believe in the plan of salvation, which addresses several of the OPs questions.

Is this Spirit Prison called Hell a place of eternal torment, or do those who have rejected Christ Jesus get a second chance to be saved?

It depends, the LDS have two definitions of hell but the spirit prison definition is not eternal. From guide to scriptures, hell

Latter-day revelation speaks of hell in at least two senses. First, it is the temporary abode in the spirit world for those who were disobedient in mortality. In this sense, hell has an end. The spirits there will be taught the gospel, and sometime following their repentance they will be resurrected to a degree of glory of which they are worthy. Those who will not repent, but are nevertheless not sons of perdition, will remain in hell throughout the Millennium. After these thousand years of torment, they will be resurrected to a telestial glory (D&amp;C 76:81–86; 88:100–101).


Second, it is the permanent location of those who are not redeemed by the Atonement of Jesus Christ. In this sense, hell is permanent. It is for those who are found “filthy still” (D&amp;C 88:35, 102). This is the place where Satan, his angels, and the sons of perdition—those who have denied the Son after the Father has revealed Him—will dwell eternally

Scriptural basis for kingdoms of heaven solely in the New Testament are limited but they include:

Matthew 13:43 (celestial glory)


43 Then shall the righteous shine forth as the sun in the kingdom of their Father. Who hath ears to hear, let him hear.


John 14:2 (degrees of glory)


2 In my Father’s house are many mansions: if it were not so, I would have told you. I go to prepare a place for you.


1 Corinthians 15:40-42 (degrees of glory)


40 There are also celestial bodies, and bodies terrestrial: but the glory of the celestial is one, and the glory of the terrestrial is another.


41 There is one glory of the sun, and another glory of the moon, and another glory of the stars: for one star differeth from another star in glory.


42 So also is the resurrection of the dead. It is sown in corruption; it is raised in incorruption:


2 Corinthians 12:2 (celestial glory)


2 I knew a man in Christ above fourteen years ago, (whether in the body, I cannot tell; or whether out of the body, I cannot tell: God knoweth;) such an one caught up to the third heaven.

Other scriptures/details about kingdoms of glory can be found in True to the Faith: Kingdoms of Glory
all emphasis added by me