According to Catholicism, does God want all humans to be Catholic? And if everyone became Catholic, would Satan be abolished? Is this acknowledged in the Bible or elsewhere?
Does God desire that all men be Catholic?
The answer to the O.P. is fundamentally “yes:” the Church teaches that it is the Catholic Church—that is, the universal Church—and hence that God calls all people to be members.
Lumen Gentium [LG], Vatican II’s dogmatic constitution on the Church, puts it as follows:
All men are called to belong to the new people of God [i.e., the Church]. Wherefore this people, while remaining one and only one, is to be spread throughout the whole world and must exist in all ages, so that the decree of God's will may be fulfilled…. It follows that though there are many nations there is but one people of God, which takes its citizens from every race, making them citizens of a kingdom which is of a heavenly rather than of an earthly nature…. All men are called to be part of this catholic [i.e., universal] unity of the people of God which in promoting universal peace presages it (LG 13).
By stating this, the Catholic Church does not intend in any way to belittle the authentic goodness in persons who are not fully in communion with the Church—that is, non-Catholic Christians, and even the non-baptized who seek the truth sincerely.
Briefly, fully-fledged members of the Catholic Church have access to all of the means of salvation that Christ gave to his Church:
They are fully incorporated in the society of the Church who, possessing the Spirit of Christ accept her entire system and all the means of salvation given to her, and are united with her as part of her visible bodily structure and through her with Christ, who rules her through the Supreme Pontiff and the bishops (LG 14).
However, other Christians enjoy a certain communion with the Catholic Church:
For men who believe in Christ and have been truly baptized are in communion with the Catholic Church even though this communion is imperfect (Vatican II’s decree on ecumenism, Unitatis redintegratio, 3).
Moreover, the Church recognizes a certain unity with those non-baptized persons who nevertheless believe in God (especially the Jewish People, as well as Muslims), and even with members of other religious who nevertheless seek the truth. (See Catechism of the Catholic Church [CCC], Nos. 839-845.)
Even so, all men are ordered to the Catholic faith:
And there belong to or are related to (ordinantur)* it in various ways, the Catholic faithful, all who believe in Christ, and indeed the whole of mankind, for all men are called by the grace of God to salvation (LG 15).
It is certainly the design of the Creator that all men should be members of his Church, while on earth; in any event, all men who are saved will be members of that Church in Heaven. (See CCC 954.)
Will Satan be banish once that is accomplished
As far as the other parts of the O.P.’s question, Satan will not be definitively “banished” or defeated until the end of the world, because, of course, the Evil One is still able to tempt even the followers of Jesus while they are on earth. This idea is expressed, for example, in Revelation 20:10:
and the devil who had deceived them was thrown into the lake of fire and sulfur where the beast and the false prophet were, and they will be tormented day and night forever and ever (ESV).
(Note that this defeat takes place immediately after the Millennium, which, according to most Catholic exegetes, refers to the age of the Church; i.e., the age we are in now, between the First and Second Coming of Christ.)
Some Biblical texts to support the Catholic view
Regarding the Biblical basis for God’s desire that all men be Catholic, there are a number of passages that could be recalled. For example, there is Acts 2:47:
And the Lord added to their number day by day those who were being saved.
The Church considers itself to be in historic continuity with the Church of the Apostles. This passage shows that God considers membership in the Church as necessary for salvation.**
If we then consider 1 Timothy 2:4, we see that God desires that all men be saved:
[God] desires all people to be saved and to come to the knowledge of the truth (ESV).
It follows that God wants all men to have access to the means of salvation, which is the Church.
In a similar vein, we have Romans 8:30:
And those whom he predestined he also called, and those whom he called he also justified, and those whom he justified he also glorified (ESV).
In Catholic thought, “predestination” is precisely the ordering by God of His creatures to Himself (i.e. His desire that we be saved).*** St. Paul affirms that being predestined also entails being called (ἐκάλεσεν), which is precisely what it means to be a member of the Church (ἐκκλησία, which even comes from the same root as “to call” in Greek).
* The official translation reads “related to,” as can be seen, but “ordered to” (that is, directed toward) would better represent the original Latin ordinantur.
** It is important, however, not to understand the Catholic Church’s necessity for salvation so extremely as to fall into rigorism. The Church does indeed teach that “outside the Church there is no salvation” (see CCC 846), but the Church also insists that those who are outside the visible confines of the Church through no fault of their own still have the opportunity for salvation.
*** Generally, however, in Catholic theology—which, at least in the West, follows St. Augustine’s terminology—the “predestined” refers to those whom God, in his omniscience, knows will be saved, whereas the “reprobate” refers to those will not be saved. This should not be taken to mean that God forces people to be saved or condemned; God always respects our freedom. The Calvinistic idea that God predestines people to Hell is quite foreign to Catholic teaching.