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 Son7., 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.
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.