Answer in brief

Do Catholics also have Assurance of Salvation?

No, Catholics have no certainty of their personal salvation, but they do have 1) an assurance that God intends to save humanity through his church, and 2) a hope that they may be among the saved.

At what point in their life can they say that they are saved and will go to Heaven?

When they have been faithful, they may be more assured of their salvation.

Or, do they strive their whole life in order to have Salvation?

Yes, Catholics believe one can never know for sure who is and is not saved, including themselves. They believe one must continue to be faithful and obedient in the hope that one day they will be saved.

Canonical source

The sixth session of the Council of Trent condemned those claiming an assurance of faith:

It is not to be said to any one boasting a confidence and certainty of the forgiveness of his sins, that his sins are forgiven, or have been forgiven; seeing this vain confidence, totally remote from piety, may exist in heretics and schismatics. ... As no pious man ought to doubt of the mercy of God, the merit of Christ, and the virtue and efficacy of the sacraments, so every one, while he beholds his own weakness and disinclination, may be in fear and dread respecting his own gracious state; seeing that no man can know with a certainty of faith, as to which there can be no lurking error, that he has obtained the grace of God. (Chapter 9)

Whosoever shall say that he holds it absolutely and infallibly certain that he shall have the great gift of perseverance even unto the end, if he has not learned this by special revelation, let him be anathema. (Canon 16)

Such people are guilty of the sin of presumption. On the other hand, one should hope in his own salvation, provided he has done good works:

Whosoever shall say that the righteous ought not, for the good works which may have been done in God, to expect and hope for eternal recompense from God through his mercy and the merits of Jesus Christ, if he persevere even to the end in well-doing and in keeping the Divine commandments, let him be anathema. (Canon 26)

Biblical basis against assurance

Ludwig Ott, in Fundamentals of Catholic Dogma (page 244), puts forward these verses as evidence in favor of the Council of Trent's declaration "against Calvin, that certainty in regard to one's predestination can be attained by special Revelation only":

Phil 2:12 Work out your salvation with fear and trembling. (cf. Ex 15:16; Jdt 2:28; Ps 2:11; Is 19:16)

I Cor 10:12 Whoever thinks he is standing secure should take care not to fall.

Other Catholics point out these verses:

I Cor 4:1-5 Thus should one regard us: as servants of Christ and stewards of the mysteries of God. Now it is of course required of stewards that they be found trustworthy. It does not concern me in the least that I be judged by you or any human tribunal; I do not even pass judgment on myself; I am not conscious of anything against me, but I do not thereby stand acquitted; the one who judges me is the Lord. Therefore, do not make any judgment before the appointed time, until the Lord comes.

The "special revelation" exception

Trent did not rule out that some might know by special revelation that they will be saved in the end.

The Catholic Church has ruled Lucia Santos' visions of Our Lady of Fatima as "worthy of belief." In one of them, Mary told Lucia that she would be taken to heaven "soon."

God only knows

According to Ott, in the same source previously quoted, our uncertainty of our predestination does not mean that God is uncertain. God knows, and has known, exactly who will be saved:

The Resolve of Predestination, as an act of the Divine knowledge and will, is as immutable as the Divine Essence Itself. The number of those who are registered in the "Book of Life" (Phil 4:3; Apoc 17:8; cf. Luke 10:20) is formally and materially fixed, that is, God knows and determines with infallible certainty in advance, how many and which men will be saved. What the number of the predestined is, God alone knows.

Perseverance to the end

Our uncertainty also doesn't mean that we can't be reasonably sure of our salvation; we just can't have an infallible assurance. After all, "The one who perseveres to the end will be saved." Matt 24:13 Ott lists a few things a person can do to be more sure of his salvation:

In spite of this uncertainty there are signs of Predetermination (signa praedestinationis) which indicate a high probability of one's predestination, e.g., a persevering practice of the virtues recommended in the Eight Beatitudes, frequent reception of Holy Communion, active love of one's neighbour, love for Christ and for the Church, veneration of the Mother of God.

Someone in a state of grace at their death can be said to have persevered to the end, while "the souls of those who die in a state of mortal sin descend into hell."

In 1 Corinthians 4, St. Paul writes:

But with me it is a very small thing that I should be judged by you or by any human court. I do not even judge myself. I am not aware of anything against myself, but I am not thereby acquitted. It is the Lord who judges me. Therefore do not pronounce judgment before the time, before the Lord comes, who will bring to light the things now hidden in darkness and will disclose the purposes of the heart. Then every man will receive his commendation from God.

Haydock’s Catholic Bible Commentary explains Paul’s meaning:

For I am not conscious. This great apostle of the Gentiles, though conscious to himself of no breach of duty, still does not dare to call himself just. How different is the conduct of this apostle, from those wicked impostors, who teach, that a man is justified by believing himself so. (Estius) — If this privileged apostle was afraid to form any judgment of his own heart and thoughts, whether they were pure or not, but left the trial thereof to the day of judgment, the day of his death, how presumptuous are they, who dare to pronounce on their election and predestination!

Rather than declare himself justified before the Lord and therefore certain of his eternal salvation, St. Paul is content to persevere in his mission and leave to God the judgment concerning his fate.

This doesn’t mean Catholics are not confident that God keeps his promises. It means simply that we must exercise caution. As our tract on salvation states, “Assurance we may have; infallible certitude we may not.”

Scripture teaches that one’s final salvation depends on the state of the soul at death. As Jesus himself tells us, "He who endures to the end will be saved" (Matt. 24:13; cf. 25:31–46). One who dies in the state of friendship with God (the state of grace) will go to heaven. The one who dies in a state of enmity and rebellion against God (the state of mortal sin) will go to hell.

It is usually argued that:

No wrong act or sinful deed can ever affect the believer’s salvation. The sinner did nothing to merit God’s grace and likewise he can do nothing to demerit grace. True, sinful conduct always lessens one’s fellowship with Christ, limits his contribution to God’s work and can result in serious disciplinary action by the Holy Spirit."

However this is not even how things work in everyday life. If another person gives us something as a grace—as a gift—and even if we did nothing to deserve it, it in no way follows that our actions are irrelevant to whether or not we keep the gift. We can lose it in all kinds of ways. We can misplace it, destroy it, give it to someone else, take it back to the store. We may even forfeit something we were given by later displeasing the one who gave it—as when a person has been appointed to a special position but is later stripped of that position on account of mismanagement.

The argument fares no better when one turns to Scripture, for one finds that Adam and Eve, who received God’s grace in a manner just as unmerited as anyone today, most definitely did demerit it—and lost grace not only for themselves but for us as well (cf. also Rom. 11:17-24). While the idea that what is received without merit cannot be lost by demerit may have a kind of poetic charm for some, it does not stand up when compared with the way things really work—either in the everyday world or in the Bible.

Whether Christians have an "absolute" assurance of salvation, regardless of their actions, consider this what Paul said: "See then the kindness and the severity of God: severity toward those who have fallen, but God’s kindness to you, provided you continue in his kindness; otherwise you too will be cut off" (Rom. 11:22; see also Heb. 10:26–29, 2 Pet. 2:20–21).

No, the doctrine of assurance is contrary to Catholic teaching. In fact, it was specifically rejected by the Council of Trent:

If any one saith, that he will for certain, of an absolute and infallible certainty, have that great gift of perseverance unto the end,-unless he have learned this by special revelation; let him be anathema. (Sixth session, Canon XVI)

For Catholics, the virtues that you list (repentence, change in character, desire for God and his Word, desire to share the Gospel, etc) are not only subsequent to salvation but actually an essential part of justifying faith.

For Catholics, the destination of the person after death is determined at the point of death. This is known as the Particular Judgement (as opposed to the Last Judgement at the end of time):

Each man receives his eternal retribution in his immortal soul at the very moment of his death, in a particular judgment that refers his life to Christ: either entrance into the blessedness of heaven-through a purification or immediately, -- or immediate and everlasting damnation. (CCC §1022)