What does Classic Dispensationalism teach about the eternal state?

It seems to be assumed that Classic Dispensationalism is taught by Darby, Scofield, Ryrie, etc. While they are all within the realms of what is viewed as 'Classic Dispensationalism', it should be noted that there are certain aspects of Darby and Scofield's teachings that are not 'classic', and those aspects are precisely about "the relationship between Israel and the church in the eternal state". That seems to be the main thrust of the question, so it seems important at the outset to clarify this.

I have a booklet called "The Scofield Bible - a Candid Examination" by Albertus Pieters, and confine my quotes from it to the eschatology and doctrine of the Church that Scofield (and Darby) espouse. You may then judge for yourself whether this is 'classic dispensational' teaching, or not.

"To state Dr Scofield's eschatological position in one sentence, he is a premillenarian of the extreme "Futuristic" school, combined with a "Darbyite" conception of the church and the kingdom.

First, then, he is a premillenarian, or more correctly, a millenarian, or chiliast. That is, he expects that the Lord Jesus Christ, at His second Advent, will establish a political kingdom on this earth, and will personally, visibly rule over it. This is an expectation that has had some currency in the early church, although not exactly in the form in which Dr Scofield holds it.

The chiliasm of the early fathers had no Jewish element - only Christians were to share in the glories of that age - while Dr. Scofield's millennium is first of all a Jewish millennium. The reign of Christ on earth at that time, he thinks, will be a sitting on the throne of David, as King of Jews, literally, strictly, and politically understood. The Gentile nations that survive the judgment of Matthew 25 will have a part in it, but only as vassal states, under the ruling Jewish power... St Augustine gave to chiliasm what was practically its death blow in the church. and it did not revive to any considerable extent until after the reformation...

Dr Scofield is not only a premillennarian however, he is a "Futuristic" Darbyite premillenarian. When we call him a "Futurist," we refer to his interpretation of the Apocalypse, in which he is an adherent of the "Futurist" school, as against the "Historicists" and the "Preterists." This means that, in his opinion nothing after the third chapter of Revelation has as yet begun to be fulfilled - all awaits fulfillment within a period of seven years, at the end of this dispensation. Also, that he regards the "Beast" of chapter 13 to the end, as a personal Antichrist, who shall then rule over the world. By no means do all premillenarians agree with him in this." The Scofield Bible - a Candid Examination Albertus Pieters, pp. 18-20, Telos Book Centre Singapore, 1955 ISBN 981-00-4066-0

Note that Scofield and Derby actually promote a third advent of Christ, due to his supposed 'secret' coming for his own, with a time gap between then and coming in judgment. They believe that there will be a literal 7-year time gap between a secret rapture and the start of Armageddon which must, of necessity, teach a third coming of Christ. This is not a 'classic' view. The booklet details a few more aspects of a 'rarefied' take on such end-time interpretations. Those in the 'classic' school:

"...believed that the Apocalypse [the book of the Revelation] was the book of the Christian Church, not of the Jews. ...Thus Dr Scofield belongs to a party within a party, a premillenarian who is also a Futurist. Not only so, but he is a "Darbyite" Futurist, i.e., he teaches that there was a complete break between the Hebrew development and the Christian church, as we shall see presently.

Not all Futurists are Darbyites [giving Dr Abraham Kuyper and A. Seiss as two examples, before listing Scofield's eschatological scheme] as follows:

(1) At any time there may take place the "Rapture," the sudden noiseless, and invisible removal from the world of all true Christians, to meet the Lord in the air. Simultaneously will take place the resurrection of all the redeemed who shall have died by that time, of all the past ages.

(2) Although there is then not a single true believer left in the world, this event will have such a remarkable effect that many hitherto unbelieving, or only nominal Christians, will turn to the Lord. These form the group called "the tribulation saints." They will begin to preach "the gospel of the kingdom" which is not the old gospel of the grace of God unto salvation, but an announcement of the imminence of the "kingdom," i.e. the earthly rule of Christ. It is thus not a continuation of the Christian message, but a resumption of the preaching of John the Baptist.

(3) Immediately now appears the "Beast" of Revelation, the Antichrist, who will bear rule both in church and state, throughout the world.

(4) At about this time, also, will take place the re-gathering of Israel, including the Ten Tribes, who, Dr Scofield teaches, are still preserved somewhere as an independent unit, known to God. To these, together with those we usually call "The Jews" the land of Palestine will be restored, according to the "Palestinian Covenant."

(5) With these restored Israelites and Jews the Antichrist will make a "Seven-year covenant" for the re-building of the temple in Jerusalem, and the re-institution of the Levitical sacrifices.

(6) In the midst of the said seven-year period, i.e., after three years and a half, the Antichrist will repudiate his promise, and will demand for himself divine worship.

(7) All the "tribulation saints" and many faithful Jews not yet Christians will refuse to render such blasphemous and idolatrous worship, and they will therefore be subject to a terrible persecution called "the great tribulation."

(8) At the end of this period, all nations will come up against Jerusalem to battle, and will almost win. They will take part of the city, but a great earthquake shall cleave the mount of Olives and a remnant will flee into the cleft for safety. This is the Battle of Armageddon." (Ibid. pp 21-22)

There are seven more points but the only parts relevant to this question are that nations (not individuals) will be judged as per Matthew 25, to determine which nations will survive into the millennium, during which time the remaining Jews are to convert and to become missionaries to the rest of the world. Then a "second stage" of the previous resurrection happens, where the martyred "tribulation saints" are raised. The millennium kingdom starts by force, Christ ruling over the Jews, who bear rule over the rest of the world. At the rebuilt temple, sacrifices will recommence and Mosaic legislation and the Sermon on the Mount will be the law. Only after that 1,000 year rule on earth will come a great revolt, the resurrection of the wicked, and the last judgment.

But it is Scofield's Doctrine of the Church that puts him into the greatest conflict within the historic Christian faith. The concluding pages of the booklet quote Scofield as claiming that

"Especially is it necessary to exclude the notion - a legacy in Protestant thought from post-apostolic and Roman Catholic theology - that the Church is the true Israel, and that the New Testament foreview of the kingdom is fulfilled in the church." And then one of Dr Scofield's ardent disciples, Dr Ironside, is quoted in agreement in his writing, Mysteries of God, p. 50: "In fact, until brought to the fore through the writings and preaching of a distinguished ex-clergyman, Mr J N Darby, in the last part of the [19th] century [the doctrine taught by Dr Scofield] is scarcely to be found in a single book or sermon throughout a period of 1600 years!" He adds that this doctrine that the Christian church being entirely unrelated to the Old Testament Israel, "was unknown to the prophets, and is not referred to in any way in their predictions." (Pieters booklet pp.24-25)

"The Darbyite doctrine, briefly stated, is that there was a clean break between the historical development of Israel and the rise of the Christian Church... The establishment of such a political regime was what Jesus came to do, what He tried to do, and what He would have done if He could. This purpose was frustrated by the refusal of the Jews to accept Him as their king... [Christ's] withdrawal of the offer to establish the "kingdom" was, however, only temporary. It will be renewed shortly before the "Revelation" of Jesus Christ, i.e. during the dark days of the Antichrist, and will be accomplished in the millennial age. This is what is called the "postponed kingdom" theory... Dr Ironside puts it thus (p. 54): "The prophetic clock stopped at Calvary. Not one tick has been heard since. From the moment Jesus bowed his head and yielded up His spirit to the Father, all the glories of the kingdom spoken of by the Old Testament seers and prophets have been in abeyance."

[The Bible] has been transformed into a Jewish book, in the sense that the traditional interpretation of the Synagogue, not of the Church, must be regarded as correct." (Ibid. p. 25-26)

Enough has been quoted to show what Darby, Scofield and Ironside believed regarding the relationship between Israel and the Church. But how they will relate in the eternal state? Another book I used to sort out the futurist, premillennial, dispensational stance was good up until the last 3 chapters of Revelation. It gave 4 parallel columns to detail the 4 main schools of thought, but it had to give that up then because the whole thing becomes incredibly muddled. The author of this other book does say about the dispensational view:

“[They] believe in a special status of the nation of Israel in the redemptive work of God in the end times… Gog and Magog – The attack is against the camp of the saints who are apparently viewed as camped around the city, mobilized for her defense, and the beloved city. The city is Jerusalem. Premillennial expositors are nearly unanimous in identifying it with the earthly Jerusalem and not with the heavenly Jerusalem seen descending in chapter 21… Among futurists, and especially the dispensationalists, there is broad agreement that chapters 20 and 21 describe fairly precisely the creation of a new universe after the millennium. The focus is terrestrial, as the majority of the details point in the direction of an earthly city to which the kings and the nations bring their treasures as gifts for the worship of God… [Re. 21:1] Many take the sea symbolically as representing the nations and peoples of the Gentiles. According to this theory, only the spiritual Israel remains of all the nations that once covered the planet. [Generally, ‘the bride’ of ch. 22 is viewed as the Church, but I have no way of knowing if that and the previous 2 sentences are a futurist, premillennial, dispensational view or not.] Revelation: Four Views, Ed. Steve Gregg, pp. 458, 474,486, 488, Thomas Nelson, 1997

I could find nothing further about their view of the Church and Israel in the eternal state. Plenty is said about the eternal state, but not in relation to Israel. However, the bulk of this answer shows the theology behind that, leading up to the eternal state, with the last bit giving a few clues about that.