What is Heaven like according to Catholics? What does a person experience who is saved, after their death?

Are they going to have a bodily resurrection in the New Jerusalem, in the real world, or is it a spiritual existence? If in the real world, will their body have any relation to the body they had, including things like age, conditions like dwarfism or siamese twins etc? Will every day have a good temperature, the sun won't eventually engulf the earth, etc?

What sources do we have on what actually happens for all eternity?
Most saints, before entering heaven, will first go to purgatory according to the Catholic church. While in purgatory they are purified ("purged") until they may enter heaven. The Catechism of the Catholic Church says regarding purgatory:


  All who die in God's grace and friendship, but still imperfectly
  purified, are indeed assured of their eternal salvation; but after
  death they undergo purification, so as to achieve the holiness
  necessary to enter the joy of heaven. The Church gives the name
  Purgatory to this final purification of the elect, which is entirely
  different from the punishment of the damned.
  ...
  Those who die in God's grace and friendship imperfectly purified,
  although they are assured of their eternal salvation, undergo a
  purification after death, so as to achieve the holiness necessary to
  enter the joy of God. By virtue of the "communion of saints," the
  Church commends the dead to God's mercy and offers her prayers,
  especially the holy sacrifice of the Eucharist, on their behalf.


The Catechism of the Catholic Church says regarding heaven:


  Those who die in God’s grace and friendship and are perfectly purified
  live for ever with Christ. They are like God for ever, for they “see
  him as he is,” face to face ... The life of the blessed consists in
  the full and perfect possession of the fruits of the redemption
  accomplished by Christ. He makes partners in his heavenly
  glorification those who have believed in him and remained faithful to
  his will. Heaven is the blessed community of all who are perfectly
  incorporated into Christ. ... Scripture speaks of it in images: life,
  light, peace, wedding feast, wine of the kingdom, the Father’s house,
  the heavenly Jerusalem, paradise: “no eye has seen, nor ear heard, nor
  the heart of man conceived, what God has prepared for those who love
  him.” ... In the glory of heaven the blessed continue joyfully to
  fulfill God’s will in relation to other men and to all creation.
  Already they reign with Christ; with him “they shall reign for ever
  and ever.”


Pope John Paul II said:


  In the context of Revelation, we know that the “heaven” or “happiness”
  in which we will find ourselves is neither an abstraction nor a
  physical place in the clouds, but a living, personal relationship with
  the Holy Trinity. It is our meeting with the Father which takes place
  in the risen Christ through the communion of the Holy Spirit.
  
  It is always necessary to maintain a certain restraint in describing
  these “ultimate realities” since their depiction is always
  unsatisfactory. Today, personalist language is better suited to
  describing the state of happiness and peace we will enjoy in our
  definitive communion with God.


Catholic Answers practices this "restraint" when it says:


  We leave up to God "where" and "how" he fulfills his promise.


But the Catholic Encyclopedia describes the opinions of theologians regarding the "location" of heaven (which are just that, opinions):


  Some are of opinion that heaven is everywhere, as God is everywhere.
  According to this view the blessed can move about freely in every part
  of the universe, and still remain with God and see everywhere.
  Everywhere, too, they remain with Christ (in His sacred Humanity) and
  with the saints and the angels. For, according to the advocates of
  this opinion, the spatial distances of this world must no longer
  impede the mutual intercourse of blessed.
  
  In general, however, theologians deem more appropriate that there
  should be a special and glorious abode, in which the blessed have
  their peculiar home and where they usually abide, even though they be
  free to go about in this world. For the surroundings in the midst of
  which the blessed have their dwelling must be in accordance with their
  happy state; and the internal union of charity which joins them in
  affection must find its outward expression in community of habitation.
  At the end of the world, the earth together with the celestial bodies
  will be gloriously transformed into a part of the dwelling-place of
  the blessed (Revelation 21). Hence there seems to be no sufficient
  reason for attributing a metaphorical sense to those numerous
  utterances of the Bible which suggest a definite dwelling-place of the
  blessed. Theologians, therefore, generally hold that the heaven of the
  blessed is a special place with definite limits. Naturally, this place
  is held to exist, not within the earth, but, in accordance with the
  expressions of Scripture, without and beyond its limits. All further
  details regarding its locality are quite uncertain. The Church has
  decided nothing on this subject.


Catholic apologist Peter Kreeft has a Q&amp;A on heaven. Here are some portions relevant to your question:


  What happens in hell?
  
  Nothing.
  
  What happens in Heaven?
  
  Everything.
  
  What kind of bodies will we have in Heaven?
  
  Christians say we will have transformed bodies, but real, physical
  bodies, as Christ had after his resurrection. His body could be
  touched and could eat. Yet it could come and go as he pleased, with
  neither walls nor distance as an obstacle. It was the same body he had
  before he died, and it was recognized as such by his friends. Yet it
  was so different that at first they did not recognize him. I think
  our new resurrection body will be related to the body we have now in
  the same way that our current body is related to the body we had in
  our mothers' wombs. If a fetus saw a picture of itself at the age of
  twenty, it would at first not recognize itself, so unforeseen and
  surprisingly new would it be. Yet it is the same self, even the same
  body, now grown radically more mature.
  
  What of injuries and deformities? Will they all be removed in the resurrection body?
  
  I think not. Christ still had his wounds. But they were badges of
  glory, not suffering and sadness. I think everything—in the body, in
  the soul, and in the person's world—that was offered to God and taken
  up into the eternal kingdom will be preserved and transformed and
  glorified in Heaven: but everything that was not—everything that was
  not the work of God or of the sanctified soul but was of the world,
  the flesh, or the devil—will be left outside Heaven's gate. The
  martyrs' wounds will glow like gold, but the amputee's limb will be
  restored, and so will the brain-damaged person's intelligence. God's
  justice and mercy are perfect, and so is his style.


Bottom line: Heaven is gonna be pretty great, but no one knows exactly how except that we shall see God.