Did the atonement end on the Cross?

That is certainly how that is interpreted by a large percentage of Christian groups, and it's how the author of Hebrews interpreted it.

Hebrews 10:10 KJV By the which will we are sanctified through the offering of the body of Jesus Christ once for all.

I assume the second part of the question deals with the fact the high priest offers sacrifices for sins. In Hebrews 5:1-3 we read...

Hebrews 5:1-3 New International Version (NIV)

5 Every high priest is selected from among the people and is appointed to represent the people in matters related to God, to offer gifts and sacrifices for sins. 2 He is able to deal gently with those who are ignorant and are going astray, since he himself is subject to weakness. 3 This is why he has to offer sacrifices for his own sins, as well as for the sins of the people.

Also

Hebrews 10 1-18 NIV

Christ’s Sacrifice Once for All

10 The law is only a shadow of the good things that are coming—not the realities themselves. For this reason it can never, by the same sacrifices repeated endlessly year after year, make perfect those who draw near to worship. 2 Otherwise, would they not have stopped being offered? For the worshipers would have been cleansed once for all, and would no longer have felt guilty for their sins. 3 But those sacrifices are an annual reminder of sins. 4 It is impossible for the blood of bulls and goats to take away sins. 5 Therefore, when Christ came into the world, he said: “Sacrifice and offering you did not desire, but a body you prepared for me; 6 with burnt offerings and sin offerings you were not pleased. 7 Then I said, ‘Here I am—it is written about me in the scroll— I have come to do your will, my God.’”[a] 8 First he said, “Sacrifices and offerings, burnt offerings and sin offerings you did not desire, nor were you pleased with them”—though they were offered in accordance with the law. 9 Then he said, “Here I am, I have come to do your will.” He sets aside the first to establish the second. 10 And by that will, we have been made holy through the sacrifice of the body of Jesus Christ once for all. 11 Day after day every priest stands and performs his religious duties; again and again he offers the same sacrifices, which can never take away sins. 12 But when this priest had offered for all time one sacrifice for sins, he sat down at the right hand of God, 13 and since that time he waits for his enemies to be made his footstool. 14 For by one sacrifice he has made perfect forever those who are being made holy. 15 The Holy Spirit also testifies to us about this. First he says: 16 “This is the covenant I will make with them after that time, says the Lord. I will put my laws in their hearts, and I will write them on their minds.”[b] 17 Then he adds: “Their sins and lawless acts I will remember no more.”[c] 18 And where these have been forgiven, sacrifice for sin is no longer necessary.

This is referring to a human priest, who must sacrifice "for his own sins, as well as for the sins of the people". Jesus, as the sinless Son of God, was sinless, so He would not have to sacrifice for His own sins.

This means that these sacrifices are ordinary sacrifices offered by sinful humans. Offering animals - the best animals, spotless, presumably, if they were following the Law, but animals nonetheless.

Jesus' sacrifice was unique. It was offered by a preist who was sinless (Christ) and the offering was perfect - something unlike any other offering that ever had, or could, be offered (The sinless Son of God).

So His sacrifice only needed to be a one-time sacrifice.

And the Priestly title? His work as Priest, at least as far as sacrifice is concerned, is finished, but He still retains the title, because it describes an aspect of who He is. The author of the passage above was showing that Jesus was greater than any other High Priest.

Much more can be found here: http://www.mycrandall.ca/courses/NTIntro/Melch8.htm

Excerpt:

Finally, in Heb 7:18, the author argues that, if there is a new High Priest, there must be a setting aside (athetêtis) of the (old) commandment (regarding priesthood) because it was weak and useless (asthenês kai anôpheles) (see "fleshly commandment" in Heb 7:16). As he already indicated, a new priesthood would not have neen necessary if the old priesthood was effective. So, according to the author, the promise of a perpetual Levitical priesthood in Exod 29:9 ("And they shall have the priesthood by a perpetual statute") and 40:14 ("Their anointing will qualify them for a perpetual priesthood throughout their generations") should be interpreted to mean until the time of the eschaton, when the greater High Priest appears. The author denigrates the Law further by saying that it made nothing “perfect” (eteleiosen) (Heb 7:19a). Actually, according to the author, the Law was not intended to be eternal but only preparatory for something better and perfect. When he says that the Law did not make perfect, he means that the Law as pertaining to the priesthood and the Temple cult did not serve to make anyone acceptable to God. As a result a better hope is introduced (through a better High Priest), which allows for a new approach to God (7:19b).

As per Roman Catholics, Ministry of High Priest of Jesus is still continuing.

CCC 1410: It is Christ himself, the eternal high priest of the New Covenant who, acting through the ministry of the priests, offers the Eucharistic sacrifice. And it is the same Christ, really present under the species of bread and wine, who is the offering of the Eucharistic sacrifice.

Christ’s Sacrifice is once for all.

Hebrews 10:10

And by that will, we have been made holy through the sacrifice of the body of Jesus Christ once for all.

Catechism of Catholic Church teaches that this is how Israel understands its liberation from Egypt: Every time Passover is celebrated, the Exodus events are made present to the memory of believers so that they may conform their lives to them. This memorial is not merely the recollection of past events but the proclamation of the mighty works wrought by God for men.

In the New Testament, the memorial takes on new meaning. When the Church celebrates the Eucharist, she commemorates Christ's Passover, and it is made present the sacrifice Christ offered once for all on the cross remains ever present. In the liturgical celebration of these events, they become in a certain way present and real.

Because it is the memorial of Christ's Passover, the Eucharist is also a sacrifice. The sacrificial character of the Eucharist is manifested in the very words of institution: "This is my body which is given for you" and "This cup which is poured out for you is the New Covenant in my blood." In the Eucharist Christ gives us the very body which he gave up for us on the cross, the very blood which he "poured out for many for the forgiveness of sins. The Eucharist is thus a sacrifice because it re-presents (makes present) the sacrifice of the cross, because it is its memorial and because it applies its fruit:

[Christ], our Lord and God, was once and for all to offer himself to God the Father by his death on the altar of the cross, to accomplish there an everlasting redemption. But because his priesthood was not to end with his death, at the Last Supper "on the night when he was betrayed," [he wanted] to leave to his beloved spouse the Church a visible sacrifice (as the nature of man demands) by which the bloody sacrifice which he was to accomplish once for all on the cross would be re-presented, its memory perpetuated until the end of the world, and its salutary power be applied to the forgiveness of the sins we daily commit.

St. Augustine admirably summed up this doctrine that moves us to an ever more complete participation in our Redeemer's sacrifice which we celebrate in the Eucharist:

This wholly redeemed city, the assembly and society of the saints, is offered to God as a universal sacrifice by the high priest who in the form of a slave went so far as to offer himself for us in his Passion, to make us the Body of so great a head. . . . Such is the sacrifice of Christians: "we who are many are one Body in Christ" The Church continues to reproduce this sacrifice in the sacrament of the altar so well-known to believers wherein it is evident to them that in what she offers she herself is offered

It is by the conversion of the bread and wine into Christ's body and blood that Christ becomes present in this sacrament. The Church Fathers strongly affirmed the faith of the Church in the efficacy of the Word of Christ and of the action of the Holy Spirit to bring about this conversion. Thus St. John Chrysostom declares:

It is not man that causes the things offered to become the Body and Blood of Christ, but he who was crucified for us, Christ himself. The priest, in the role of Christ, pronounces these words, but their power and grace are God's. This is my body, he says. This word transforms the things offered.

And St. Ambrose says about this conversion:

Be convinced that this is not what nature has formed, but what the blessing has consecrated. The power of the blessing prevails over that of nature, because by the blessing nature itself is changed. . . . Could not Christ's word, which can make from nothing what did not exist, change existing things into what they were not before? It is no less a feat to give things their original nature than to change their nature.

As often as the sacrifice of the Cross by which 'Christ our Pasch has been sacrificed' is celebrated on the altar, the work of our redemption is carried out.

CCC 1348 All gather together. Christians come together in one place for the Eucharistic assembly. At its head is Christ himself, the principal agent of the Eucharist. He is high priest of the New Covenant; it is he himself who presides invisibly over every Eucharistic celebration. It is in representing him that the bishop or priest acting in the person of Christ the head (in persona Christi capitis) presides over the assembly, speaks after the readings, receives the offerings, and says the Eucharistic Prayer. All have their own active parts to play in the celebration, each in his own way: readers, those who bring up the offerings, those who give communion and the whole people whose "Amen" manifests their participation.