The Day of Atonement is referred to in the Hebrew in Leviticus as literally "day of the atonements" (plural). Yom haKippurim.
There were three things atoned for on that day:
1. The high priest and his house. (The priest had to be clean in order to clean things up, obviously).
2. The sanctuary and its furniture.
3. The people/nation.
All through the year, the people brought their various sin offerings and tresspass offerings. The animals were slain, their blood poured out at the altar, and their bodies burnt on the altar. This caused a ritual pollution of the sanctuary:
Leviticus 16:16 KJV
And he shall make an atonement for the holy place, because of the uncleanness of the children of Israel, and because of their transgressions in all their sins: and so shall he do for the tabernacle of the congregation, that remaineth among them in the midst of their uncleanness.
Leviticus 16:18-20 KJV
And he shall go out unto the altar that is before the LORD, and make an atonement for it; and shall take of the blood of the bullock, and of the blood of the goat, and put it upon the horns of the altar round about. [19] And he shall sprinkle of the blood upon it with his finger seven times, and cleanse it, and hallow it from the uncleanness of the children of Israel. [20] And when he hath made an end of reconciling the holy place, and the tabernacle of the congregation, and the altar, he shall bring the live goat:
Once the sanctuary and its furniture had been cleansed and "covered" (atoned, reconciled), then the live goat (the scapegoat) was brought forth, the sins of the people were placed upon it, and it was sent away never to be seen again.
A sin offering was made for the people, the sanctuary was then cleansed, and then the scapegoat bore the sins of the people away.
So all through the year the sins of the people were brought into the sanctuary so to speak, and their blood was deposited at the altar. So the sanctuary served as an accumulation point if you will for the sins of the people. A spiritual "septic tank" if you will. And just like a regular septic tank must be emptied periodically, the sanctuary had to be "emptied" of its pollutions (the sins of the people) periodically (once every year).
Obviously, this could never actually take away sins, as Hebrews tells us. Jesus was the one-and-done sacrifice for sins, and actually takes away our sins (not just the guilt, but the tresspassing as well) so that we can now serve the living God from the heart. As the scapegoat was sent off never to be seen again, Jesus takes our sins and removes them from us as far as the east is from the west (ie infinitely).
Now, the earthly sanctuary was a pattern, a physical mock-up of a spiritual reality, an earthly representation of heaven (the throne room/sanctuary of God). The earthly sanctuary was ritually unclean because all the sins of the people were being ritually deposited there throughout the year. The true sanctuary (the dwelling place of God) was also defiled (in a sense) by the actual sins of the people, but Christ removed all that once and for all.
What is the dwelling place or sanctuary or temple of God? We may point up to the firmament and say it some "place" on the other side of it, but is it not also true that His temple, His sanctuary, His dwelling place was always intended to be HIS PEOPLE? Is this not where things end up at the end of the Revelation? The Tabernacle was surrounded by the camps of the tribes of Israel. And the banners of the four main camps (one on each side) correspond to the four cherubim or four living creatures of both Ezekiel and Revelation. Thus, the sanctuary represents the presence of God in the midst of His people, so that heaven is essentially God and His people. We see a distinction between earth and heaven, but in the end the two will be juxtaposed and merged so that there is no difference. Heaven and earth will be one thing, and God and His people will be the ones living there, TOGETHER. The Christian walk is a picture in miniature of this truth - we have the divine heavenly Spirit of God in these "earthen vessels of clay". In Christ, the divine and human are joined in One, God and Man in one person. This is the entire PLAN of God from the beginning - the manifestation of God in the flesh, the union of human and divine, heaven-on-earth, etc.
So the heavenly sanctuary must needs be cleansed because of the sins of His people, in order for Him to dwell with them in their midst. And this cleansing was done in the death, burial, and resurrection of Jesus. His mission was to prepare the sanctuary -
John 14:1-4 KJV
Let not your heart be troubled: ye believe in God, believe also in me. [2] In my Father's house are many mansions: if it were not so, I would have told you. I go to prepare a place for you. [3] And if I go and prepare a place for you, I will come again, and receive you unto myself; that where I am, there ye may be also. [4] And whither I go ye know, and the way ye know.
What is the Father's "house"? The temple, for a temple is the "house" of a god, and the true God has His Temple. There are many "mansions" (dwelling places, abodes) in the true Temple of God. Jesus was going to prepare a place (a dwelling place) for His disciples in the Temple:
Revelation 3:12 KJV
Him that overcometh will I make a pillar in the temple of my God, and he shall go no more out: and I will write upon him the name of my God, and the name of the city of my God, which is new Jerusalem, which cometh down out of heaven from my God: and I will write upon him my new name.
The church is the temple of the living God. The church is the people of God, the holy nation, the Israel of God, the desired dwelling place of God, ZION:
Psalm 76:1-2 KJV
In Judah is God known: his name is great in Israel. [2] In Salem also is his tabernacle, and his dwelling place in Zion.
Hebrews 12:22-24 KJV
But ye are come unto mount Sion, and unto the city of the living God, the heavenly Jerusalem, and to an innumerable company of angels, [23] To the general assembly and church of the firstborn, which are written in heaven, and to God the Judge of all, and to the spirits of just men made perfect, [24] And to Jesus the mediator of the new covenant, and to the blood of sprinkling, that speaketh better things than that of Abel.
As to whether or not it is still ongoing, consider this:
Hebrews 9:24-28 KJV
For Christ is not entered into the holy places made with hands, which are the figures of the true; but into heaven itself, now to appear in the presence of God for us: [25] Nor yet that he should offer himself often, as the high priest entereth into the holy place every year with blood of others; [26] For then must he often have suffered since the foundation of the world: but now once in the end of the world hath he appeared to put away sin by the sacrifice of himself. [27] And as it is appointed unto men once to die, but after this the judgment: [28] So Christ was once offered to bear the sins of many; and unto them that look for him shall he appear the second time without sin unto salvation.
The high priest went into the sanctuary, and then came out when it was all done. Chrisrt has gone into the heavenly sanctuary, but has not yet appeared or "come out" yet. Thus, the cleansing is ongoing. Why? Because the temple is not completed, the church is still having people added to it, and total number of the elect who are to have their sins fully removed has not yet come in. So the current time period is one of Atonement, that is, Christ still removes sin, still removes guilt, still prepares a place for YOU in His holy tabernacle.
Once Atonement has been completed, Tabernacles is next on the calendar:
Revelation 21:1-4 KJV
And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea. [2] And I John saw the holy city, new Jerusalem, coming down from God out of heaven, prepared as a bride adorned for her husband. [3] And I heard a great voice out of heaven saying, Behold, the tabernacle of God is with men, and he will dwell with them, and they shall be his people, and God himself shall be with them, and be their God. [4] And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away.
So, did the blood of Christ actually drip on to the physical mercy seat? I would say NO, because that would violate the very pattern established by Scripture, in that the earthly mercy seat was just a man-made representation of the thing that Jesus' blood was intended to purify. Animal blood for the earthly sanctuary, Christ's blood for the heavenly sanctuary. To put Jesus' blood on the earthly sanctuary as an atonement would be not much different that putting goat blood on the heavenly sanctuary as an atonement. In other words, not only unnecessary, but backwards.