What is the Catholic understanding of Predestination and Freewill?

Is predestination a Catholic doctrine? If so, how does the Catholic Church harmonise the doctrine of predestination against freewill?

The simple answer is YES.

The Catholic understanding of predestination or divine election encompasses man’s free-will response in accepting God’s gift of eternal salvation. As Augustine, the great Father and Doctor of the Church, summarized so well, “God created us without us: but he did not will to save us without us” (CCC 1847).
Augustine Had It Right; Calvin Did Not

I offer two resources to help answer your second question in more detail.

Catholic blog article, which sets the stage

The 2014 Catholic blog article How to Understand Predestination from a Catholic Perspective describes the Catholic Church's understanding of both predestination and free will, which differs from the Protestant Reformed's understanding.

First, the article offers a definition by way of CCC 600 then commented:

Notice how predestination is defined. God knows all who will accept His gift of saving grace. So, for all time, the knowledge of God being unlimited, God has known whom would say “yes” to His grace. This is the plan of salvation offered to us from the Father, through Jesus, by the power of the Holy Spirit. The confusion about this doctrine arises when theologians attempt to explain how the mystery of free will, grace, and the fore-knowledge of God all fit together. There are several options for Catholics to choose from and we are free to do so. But, there are also some positions the Church rejects.

Then the article mentions the two options rejected by the Catholic Church:

1. Pelagianism which teaches that we don't need God's assistance to save ourselves. This option was refuted by St. Augustine then by the 431 AD Council of Ephesus.
2. Double predestination from the Calvinist/Reformed tradition which teaches that God actively chooses some to go to heaven and damns others to hell. Although Reformed tradition claims support of St. Augustine, the Catholic Church has always rejected this option. See the Catholic.com article Augustine Had It Right; Calvin Did Not.

The article helpfully places the Catholic doctrine as between the two extremes represented by the two options:

Notice the first error removes God as the prime actor and second removes any cooperation from man.

The article then mentions two camps which try to define the middle ground more fully:

1. By St. Augustine which St. Thomas Aquinas expanded. This camp emphasizes grace.
2. By Molina and other scholars. This camp emphasizes free will.

Neither camp rejects other side's emphasis. A Catholic has the freedom to choose how to work it out within the middle ground.

The article then finishes with some connections with other doctrines to guide the faithful in working out the middle ground, which also answers your second question: "how does the Catholic Church harmonise the doctrine of predestination against freewill":

• God is the source of all good. God does not create evil (which isn’t really a “thing”, but rather an absence of a good). God cannot do an evil act.
• God allows humans to choose to do good or evil. We have free will. It is possible to reject God’s grace.
• God’s knowledge is infinite. There is nothing He does not know.
• God wills (desires) that all be saved.
• God always acts first. His grace comes and then we are empowered by it to be able to respond.
• Even after saving grace is received, we can reject it later.

Catholic thesis for MA in theology, which develops one way to answer

One example how a Catholic may develop this middle ground (by way of St. Thomas Aquinas), is the excellent 2013 thesis you quoted: An Historical and Theological Survey of the Catholic Doctrine of Predestination by Andrew J. Allen for his Master of Arts in Theology at Saint Paul Seminary School of Divinity (University of St. Thomas, Minnesota).

The thesis has 2 goals:

1. Provide an historical survey of the development of the Catholic doctrine of predestination to uncover its presence in the Tradition
2. Present the theological content of the doctrine with an emphasis on the teaching of St. Thomas Aquinas, which we believe to be a faithful theocentric account according to Sacred Scripture and Tradition, arguing against the noted theological errors about the nature of God and free will.⁷ For St. Thomas never tires of asserting God’s primacy and yet upholds man’s dignity. For him, all glory given to creatures is glory that redounds to God.

Excerpts from the concluding chapter of the thesis (emphasis mine):

...

It will be helpful to highlight the most important distinctive features of Aquinas' position, and briefly explain why we think they are satisfying. Following Garrigou-Lagrange, we see that Thomas effectively affirmed two great biblical principles relating to the doctrine of predestination which cannot be separated without negative theological consequences:

1. The principle of predilection (which refutes the error of pelagianism by asserting God’s causal primacy with regard to predestination and grace)
2. That God is infinitely just, implying the integrity of human freedom and that God never commands what is impossible (this refuting the error of predestinarianism [The Calvinist/Reformed's understanding]).

While the two are approachable separately, their reconciliation is the impenetrable mystery that continues to evade the grasp of human thought.

St. Thomas emphatically states that the prevision of merit is not the cause of predestination, and that “there is no reason, except the divine will,”²⁷⁵ which is perfect goodness in union with divine wisdom, and therefore just. ... While the act of the divine will is one and unchanging, and thus God loves all equally, it is the case that there are varying degrees of goodness willed. St. Thomas’ theological argument for predilection is in response to the question of whether God chooses the elect:

Predestination…is a part of providence. Now providence as also prudence, is the plan existing in the intellect directing the ordering of some things towards an end…But nothing is directed towards an end unless the will for that end already exists. Whence the predestination of some to eternal salvation presupposes, in the order of reason, that God wills their salvation; and to this belong both election and love: love, inasmuch as He wills them this particular good of eternal salvation; since to love is to wish well to anyone…election, inasmuch as He wills this good to some in preference to others; since He reprobates some.
_{Summa theol. I, q. 23, a. 4}

Unlike us, God first loves (as the cause of goodness in creatures) then elects (based on the good he has caused), then predestines. ... And so, according to Sacred Scripture and theological conclusion we must accept that the cause of predestination is found in the goodness of God. It will be worth quoting St. Thomas at length when he argues against the proposition: “God does not prepare unequal things for men by predestinating and reprobating, unless through the foreknowledge of their merits and demerits:” [Summa theol. I, q. 23, a.5, ad 3]

The last part of St. Thomas’ description, and the statements from the Council of Valence above, are important in defining reprobation in a Catholic sense by making a clear distinction. God’s will in reprobation is permissive as regards sin, of which man is formally the cause. God provides only for the material aspect of sin. And he only positively reprobates men. Unlike the Reformers, and specifically John Calvin, the Church has never taught that God positively reprobates men to the eternal punishment of hell antecedent to the consideration of demerits. This would contradict the great biblical affirmation that God wills the salvation of all men,[1 Tim 2:4] the free will of man, and the very redemption offered by Christ. We also can recall the litany of Eastern Fathers that cited the cause of sin to be the free will of man, and that the sin is justly deserving of the punishment willed by God. As St. John Damascene said,

The total desertion happens when, after God has done everything to save, the man remains unreformed and not cured, or rather, incurable, as a result of his own resolve. Then he is given over to complete destruction, like Judas…For He did not make us to punish, but to share in His goodness, because He is good. But He wills that sinners be punished, because he is just.
_{St. John Damascene, De fide orthodoxa 2.29, quoted in Most, Grace Predestination and the Salvific Will of God, ²⁷²}

It will be worth another look at St. Thomas’ description of the cause of reprobation:

Reprobation differs in its causality from predestination. This latter is the cause both of what is expected in the future life by the predestined—namely, glory—and of what is received in this life—namely, grace. Reprobation, however, is not the cause of what is in the present—namely, sin; but it is the cause of abandonment by God. It is the cause, however, of what is assigned in the future—namely, eternal punishment. But guilt proceeds from the free will of the person who is reprobated and deserted by grace. [Summa theol., I, q. 23, a. 3, ad 2um]

... As we have seen in chapter 1, the Scholastic theologians following St. John Damascene explain how God’s will of salvation is truly for all by distinguishing between the antecedent and consequent will. The former is directed to what is good in itself regardless of the circumstances, thus God wills antecedently that all men are saved. However, as to the actual circumstances, God wills subsequently that some are damned on account of the divine permission that they fall into and remain in sin. He gives truly sufficient graces to all with the real possibility of following his commandments. Also, adopting a Christocentric emphasis, we must see in Jesus Christ the definitive desire of God: the salvation of all through divine kenosis. An emphasis on the election of Christ as the interpretive key in our doctrine shows that even though God is transcendent and mysterious, that his revelation in Christ is indicative of the profound love of God for humanity.

... In the introduction of this paper it is said that there are two underlying theological errors concerning the doctrine of predestination prevalent in our day: that God is just another inhabitant of the universe, and that human freedom requires radical independence from God. Concomitant to these errors is the conviction that predestination is a Reformed doctrine that should be avoided as an anomalous development in the Christian theological tradition. As to this last point, even in this modest historical and theological survey, it is shown that the doctrine is an object of faith in Catholic teaching, and the object of continual theological investigation in the Catholic tradition. In other words, resting on the shoulders of Catholic Tradition, there is not only evidence that predestination should be studied, but preached, taught, and believed.

Conclusion (in my own paraphrase)

In trying to summarize the thesis above in one paragraph of simple language for us non scholars, I run the risk of oversimplifying what the thesis itself already warns that at the core there is an "impenetrable mystery that continues to evade the grasp of human thought." I just need to steer clear of the two errors already described above. Here's my attempt:

God, out of His goodness, wisdom, justice, and foreknowledge of the choices we make throughout our life time (God can do this because He is outside time) provides all the grace that the elects need to accept His gift of salvation and also to empower our free will to choose and do good. In Aquinas's scheme God first loves (as the cause of goodness in creatures), then elects (based on the good he has caused), then predestines. However, God does NOT predestine people to hell, but allows people to freely reject God's gift of salvation, choose sin and do evil, which is the cause of God's abandonment. In a way, God ratifies our rejection and punish us accordingly as a reprobate. Therefore, within our subjective time horizon and without our awareness of how God infuses grace to our soul it is our fearsome responsibility to accept God's offer of salvation and then to use our free will to continually choose good and do good (following God's commandments) while asking God for the grace we need (especially when we fall into sin) so we will finally arrive safely in heaven after we die.