What is Heaven like according to Catholics?

As for bodily imperfections that are a result of original sin (dwarfism,illnesses,old age wrinkles,genetic imperfections etc) They will not be part of the resurrected glorified body.

Opening

Let's start here: Before the world ever was, the Son, God, tells us that he was with the Father, God, in Glory. ¹ This Being God [the One who IS] is, according to Church teaching, "most happy in and by Himself."²

God the Father tells us who it is in whom is ALL his delight: His Son, the Beloved.² So he who in whom is ALL of God's delight must be the highest Good there is.

The Son, who the Father in his love has given to us⁴, in turn asks the Father to send his promise, the Holy Spirit,⁵ God, altissima donum Dei⁶ (the highest of God's gifts), who molds us in the image of the Son, so that through the Son and with the Son and in the Son^7., we have access to the Father⁸, on whose face the angels gaze⁹.

Again, what is the highest Good? It is a who: The trinity in Unity and the Unity in Trinity.¹⁰

Cf. CCC 257 (sans references) "O blessed light, O Trinity and first Unity!" God is eternal blessedness, undying life, unfading light. God is love: Father, Son and Holy Spirit. God freely wills to communicate the glory of his blessed life. Such is the "plan of his loving kindness", conceived by the Father before the foundation of the world, in his beloved Son: "He destined us in love to be his sons" and "to be conformed to the image of his Son", through "the spirit of sonship". This plan is a "grace [which] was given to us in Christ Jesus before the ages began", stemming immediately from Trinitarian love. It unfolds in the work of creation, the whole history of salvation after the fall, and the missions of the Son and the Spirit, which are continued in the mission of the Church.

We know that everything there is has a purpose, an end for which it was made. Therefore a thing is "happy" when it attains its purpose. The Church teaches that we are "the only creature[s] on earth that God has willed for its own sake" ¹¹. We have been made for God [cf. CCC 1700] and therefore even in this life, apart from God, ours will be the experience and hopefully, the eventual realization of St. Augustine:

[For] You have formed us for Yourself, and our hearts are restless till they find rest in You. - Source: CHURCH FATHERS: Confessions, Book I (St. Augustine) | New Advent.

God willing or God forbid the realization comes in the next life:

“All your life an unattainable ecstasy has hovered just beyond the grasp of your consciousness. The day is coming when you will wake to find, beyond all hope, that you have attained it, or else, that it was within your reach and you have lost it forever.” - C.S. Lewis, The Problem of Pain.

Answering

What is Heaven like according to Catholics?

It is where its citizens will achieve Christian Beatitude [cf. CCC 1720 ff]. Please see also the opening above summarized below:

CCC 1721 God put us in the world to know, to love, and to serve him, and so to come to paradise. Beatitude makes us "partakers of the divine nature" and of eternal life.¹² With beatitude, man enters into the glory of Christ¹³ and into the joy of the Trinitarian life.

What exactly is this communion with God like? We have been told that we have yet to experience anything like it:

CCC 1027 This mystery of blessed communion with God and all who are in Christ is beyond all understanding and description. 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."¹⁴

From above, in the communion with God, there is also a blessed communion with all who are in Christ. In the first place Holy Mary, His Mother, our Mother. Heaven is where God's saying [to the blessed elect] Son, you are always with me, and all that is mine is yours is finally fulfilled never again to be lost.

What does a person experience who is saved, after their death?

They like everyone else appear before the judgment seat of Christ, and the saved go to heaven [see above] directly or via purgatory.

Are they going to have a bodily resurrection in the New Jerusalem, in the real world, or is it a spiritual existence?

Yes. [The] resurrection of the body is an article of the Faith.

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?

2 Beloved, we are God’s children now; it does not yet appear what we shall be, but we know that when he appears we shall be like him, for we shall see him as he is. - [1 John 3:2 (RSVCE)].

In the first place the blessed look like God and in the second place like Jesus, God-made-man. Scripture tells us what in his body he was able to do after the Resurrection.^c Please see also Characteristics of the risen body | General Resurrection | New Advent: the bodies of the saints shall be distinguished by four transcendent endowments, often called qualities: "impassibility;" "brightness," or "glory;" "agility;" and "subtility."

^{c. Commentary: Jesus kept his wounds. Victorious soldiers value and look upon their scars of war on their bodies with pride. I would want to keep as much as I will be allowed to keep - they won't hinder in any way whatsoever a glorious body. After all, it is through this body, by his grace, I did achieve victory over my enemies - another cause for rejoicing in heaven.}

^{6 With a freewill offering I will sacrifice to thee; I will give thanks to thy name, O Lord, for it is good. 7 For thou hast delivered me from every trouble, and my eye has looked in triumph on my enemies. - Psalm 54 (RSVCE), Prayer for Vindication.}

Will every day have a good temperature, the sun won't eventually engulf the earth, etc?

There will be a mysterious renewal which scripture calls "new heavens and a new earth." The universe itself will be renewed and be illumined by God's glory. [Cf. Rev 21:1 (RSVCE) footnote and CCC 1042 - 1050].

When the sun will die belongs to science [cf. This SPACE.com animation]. Whether this shall happen before the renewal, only God knows. I personally believe the LORD is at hand.

What sources do we have on what actually happens for all eternity?

You may wish to start with the answers here.

^{1. Cf. John 17:5 (RSVCE).}

^{2. Cf. The Nature and Attributes of God | New Advent.}

^{3. Cf. Mark 1:11 (RSVCE.}

^{4. Cf. John 3:16 (RSVCE).}

^{5. Cf. John 14:15-17 (RSVCE).}

^{6. Cf. Veni, Creator Spiritus (Come Holy Spirit, Creator Blest) | EWTN.}

^{7. Cf. The concluding doxology, Eucharistic Prayer.}

^{8. Eph 2:18 (RSVCE).}

^{9. Matt 18:10 (RSVCE).}

^{10. Cf. Quicumque.}

^{11. Cf. CCC 356.}

^{12. 2 Pet 1:4; cf. Jn 17:3}

13. Cf. Rom 8:18

14. 1 Cor 2:9.

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&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.