The themes are actually handled no different by Edwards or Reformed Theologians then they are by other Christian denominantions. Theologians generally do not mean that God has no feelings when saying that he is 'unchanging in his perfect boundless joy'. From the standpoint of impassibility, Edwards is not opposing the same concept detailed by Thomas Aquinas (or many other earlier Christian theologians).

Immutability (Unchangeability). In his epic Summa Theologica (1a.9.1), Aquinas offers three basic arguments in favor of God’s unchangeability. The first argument is passed on the fact that a God of Pure Actuality (“I-Am-ness”) has no potentiality. It follows, therefore, that God cannot change (Exod. 3:14). Whatever changes has to have the potential to change. But as pure Actuality, God has no potential, so he cannot change.

The second argument for God’s immutability follows from his simplicity. Everything that changes is composed of what changes and what does not change. God cannot change because an absolutely simple being has no composition. If everything about a being changed, then it would be an entirely new being. In fact, it would not be change but annihilation of one thing and a creation of something entirely new. Now if, in every change in a being something remains the same and something does not, then it must be composed of these two elements. So an absolutely simple Being with no composition cannot change.

The third argument for God’s unchangeability argues from his absolute perfection. Whatever changes acquires something new. But God cannot acquire anything new, since he could not be better or more complete. Therefore, God cannot change. If he did, he would not be God for he would have lacked some perfection.

Aquinas also argues that God alone is immutable (Summa Theologica, 1a.9.2). All creatures exist only because of the will of the Creator. His power brought them into existence, and it is his power that keeps them in existence. Therefore, if he withdrew his power they would cease to exist. Whatever can cease to exist is not immutable. Therefore, God alone is immutable; everything else could cease to exist.

Impassability (without Passions).

A long-recognized attribute of God that has recently come under attack is impassability. God is without passions. Passion implies desire for what one does not have. But God, as an absolutely perfect Being, lacks nothing. To lack something he would have to have a potentiality to have it. But God is Pure Actuality with no potentiality whatsoever. Therefore, God is completely and infinitely satisfied in his own perfection.

However, to say that God is impassable in the sense that he has no passions or cravings for fulfillment is not to say that he has no feelings. God feels anger at sin and rejoices in righteousness. But God’s feelings are unchanging. He always, unchangingly, feels the same sense of anger at sin. He never ceases to rejoice in goodness and rightness. Thus, God has no changing passions, but he does have unchanging feelings. (Baker Encyclopedia of Christian Apologetics, Geisler, Norman p283)

Actually Edwards in the same book that you quote from dispells the notion that you thought he meant: God is without emotions. You are asking about a negative emotion like 'suffering'. Here Edwards is arguing how God can have a positive emotion like 'joy', in response to his creation's own actings in time.

Many have wrong notions of God’s happiness, as resulting from his absolute self-sufficience, independence, and immutability. Though it be true, that God’s glory and happiness are in and of himself, are infinite and cannot be added to, and unchangeable, for the whole and every part of which he is perfectly independent of the creature; yet it does not hence follow, nor is it true, that God has no real and proper delight, pleasure, or happiness, in any of his acts or communications relative to the creature, or effects he produces in them; or in any thing he sees in the creature’s qualifications, dispositions, actions and state.

God may have a real and proper pleasure or happiness in seeing the happy state of the creature; yet this may not be different from his delight in himself; being a delight in his own infinite goodness; or the exercise of that glorious propensity of his nature to diffuse and communicate himself, and so gratifying this inclination of his own heart. This delight which God has in his creature’s happiness, cannot properly be said to be what God receives from the creature. For it is only the effect of his own work in and communications to the creature; in making it, and admitting it to a participation of his fulness. As the sun receives nothing from the jewel that receives its light, and shines only by a participation of its brightness. (The Works of Johnathan Edwards, Volum1 1, p102).

The point is around the subject of having a chnage by being acted upon. We act in time. God acts only in eternity with all his boundless attributes. For example, we do things that God always hates and therefore expresses displeasure in, or suffers over it. However his communication of displeasure in time, is actually his unchangind displeasure from eternity, outside of time. God suffered over the sins of man before th ecreation of the world, knowing the end from the beggining. Therefore his suffering does not imply an actualy change in God. He has alwasy had displeasure towards sin and has always had empathy for those who suffer in time. Clearly we see his empathy in the nature of Christ.

So how would Edwards answer your question? He would not disagree with your own sentiments as far as I can tell. He would say what he has already said: (I here re-quote Edwards but switch the emotion from a positive example, to a negative one by adding CAPITALS where I have edited the original)

God may have a real and proper DIS-pleasure or WOULD SUFFER in seeing the UN-happy state of the creature; yet this may not be different from his delight in himself; being a delight in his own infinite goodness (AS IT IS FROM HIS GOODNEESS THAT HE HATES EVIL); or the exercise of that glorious propensity of his nature to diffuse and communicate himself, and so gratifying this inclination of his own heart. This HATRED which God has in his creature’s UN-happiness, cannot properly be said to be what God receives from the creature. For it is only the effect of his own work in and communications to the creature; in making it, and admitting it to a participation of his fulness AND HATING ITS OWN SELF-HARM BY ITS REBELLION. As the sun receives nothing from the jewel that receives its light, and shines only by a participation of its brightness.

However we should keep in mind that 'joy' not 'sorrow' is associated with God overall. The saints in heaven are not crying. Joy is God's nature. Therefore, what Edwards is arguing in the first point is that from eternity when God had all his joys and suffering expressed in everything that would happen in time, before he created the material world, his overall reason for creating the world was good to him. God can't get depressed about what he decided to do when creating the world, for it was according to his perfect will that he took infinite pleasure in creating the world, knowing all the joys and sorrows that this decision would create. His expression if himself in the creation, though with its own flaws due to sin, is still a worthy expression of which he has joy, otherwise he would have not created it. The reason for creating the world must be derived in God's own pleasure, that is what the book is about. The time based 'suffering' that God communicates, in his glorious love to his creation when they sin or are inflicted by the evil that sin has made, is based on his infinite hatred of sin. God takes infinite joy in hating sin abd suffering with it and working out those evils to the highest good in eternity. Even Christ the God-Man 'for the joy set before him', endured the cross. This is the way refomed theologinas think about these themes.

The point Edwards makes here is that since God accomplishes his purposes in all things, even evil things, he cannot be said to be carried away by emotions and thus experience involuntary ecstasy or distress. But Edwards clearly admits that God can feel pleasure – the crucial point being that God, not the creature, is its ultimate source:

Though he has real pleasure in the creature's holiness and happiness; yet this is not properly any pleasure which he receives from the creature. For these things are what he gives the creature.¹

Similarly, Edwards refers to God's "hatred"² and "indignation"³ toward sin, but contends that since he wills that it occur, and that it ultimately glorifies him, it does not distress him:

There is no inconsistence in supposing, that God may hate a thing as it is in itself, and considered simply as evil, and yet that it may be his Will it should come to pass, considering all consequences.⁴

The judgments God executes on the wicked, are spoken of as being for the sake of his name, in like manner as for his glory.⁵

This is often referred to as the doctrine of impassibility, that is, that God does not have passions. J. I. Packer explains:

Not impassivity, unconcern, and impersonal detachment in face of the creation; not insensitivity and indifference to the distresses of a fallen world; not inability or unwillingness to empathize with human pain and grief; but simply that God's experiences do not come upon him as ours come upon us, for his are foreknown, willed and chosen by himself, and are not involuntary surprises forced on him from outside, apart from his own decision, in the way that ours regularly are.⁶

In this light, some passages in which God is said to experience negative emotions (grief, jealousy, etc.) are often said to be anthropomorophisms. Millard Erickson argues that stories of God experiencing pain "are simply descriptions of God's actions and feelings in human terms, and from a human perspective,"⁷ and A. A. Hodge writes:

When [the Scriptures] speak of his repenting, of his being grieved, or jealous, they use metaphorical language also, teaching us that he acts toward us as a man would when agitated by such passions. Such metaphors are characteristic rather of the Old than of the New Testament, and occur for the most part in highly rhetorical passages of the poetical and prophetical books.⁸

(Specifically on the similar issue of God "repenting," see How have Calvinist theologians dealt with statements about God repenting?)

Many in the Reformed tradition desire to carefully safeguard the idea that God can experience emotions. For example, John Frame argues that scriptural references to God's compassion, delight, jealousy, and grief are examples of emotions, and that emotion is inseparable from intellect and will. Like Edwards, he appeals to God's ordaining will to reconcile God's emotional responses with his impassibility:

God responds (both transcendently and immanently) only to what he has himself ordained. He has chosen to create a world that often grieves him.⁹

That said, some Reformed thinkers, while still accepting the immutability of God, don't find the doctrine of impassibility helpful. For example, Wayne Grudem:

I have not affirmed God's impassibility in this book. Instead, quite the opposite is true, for God, who is the origin of our emotions and who created our emotions, certainly does feel emotions.¹⁰

Summary

Some recent Reformed theologians are less committed to the traditional doctrine of God's impassibility, suggesting that it does not align with scriptural references to God's pity, grief, and regret. But Jonathan Edwards and the general Reformed tradition tend to interpret such passages as anthropomorphic or analogical – not in order to deny that God is relational, but to argue that he is never carried away by passions or altered in any way by his creatures. As Richard Muller summarizes early Reformed thought:

In the usages found in the Christian tradition, immutability (or, indeed, impassibility, when the term is actually used) in no way implies an absence of relatedness, love, long-suffering, compassion, mercy, and so forth. Impassibility, when attributed to God in the Christian tradition and, specifically, in medieval and Protestant scholastic throught, indicates, not a Stoic notion of apatheia, but an absence of mutation, distress, or any other sort of negative passiones.¹¹

References:

1. Edwards, Works, v2, 212
2. Ibid., 197
3. Ibid., 380
4. Ibid., 162
5. Ibid., 238; emphasis in original
6. Packer, "Theism for Our Time," quoted in Robert Reymond, Systematic Theology, Chapter 7
7. Erickson, Systematic Theology, Chapter 11
8. Hodge, The Westminster Confession of Faith: A Commentary, Chapter 2, 74
9. Frame, Systematic Theology, 412–14; emphasis in original
10. Grudem, Systematic Theology, 166
11. Muller, Post-Reformation Reformed Dogmatics, v3, 310; emphasis in original