Sort of. Yes to Gentile proselytes, but the Mosaic Covenant was a spin-off from the Abrahamic covenant where God said He would make Abraham a great nation. It was a promise of an inheritance for his posterity.
Not exactly.
Romans 9:7-8
King James Version (KJV)
7 Neither, because they are the seed of Abraham, are they all children: but, In Isaac shall thy seed be called.
8 That is, They which are the children of the flesh, these are not the children of God: but the children of the promise are counted for the seed.
Israel is made distinct from Abraham's other offspring. Israel is the one brought under the Covenant of Law. But...this does not nullify God's promise to Abraham. All families of the earth were to be blessed through him, and the Seed in view was Christ Himself. The Covenant of Law provided a means of relationship to both Jew and Gentile, and the eventual fulfillment of the promises are found in the establishing of the New Covenant, which might be seen using the terminology you use above, a "spin-off" from the preceding Covenants.
For example, when God made promise to Abraham concerning all families of the earth, that promise is fulfilled in the New Covenant.
When God made promise through Isaiah's Prophecy (that Christ would die for us [Isaiah 53], that He would be made a Covenant unto the Gentiles Isaiah 42]) it is fulfilled in the New Covenant.
I take the basic view that the New Covenant is progressively revealed to men throughout the course of Redemptive History as recorded in Scripture, beginning with Genesis 3:15.
The Old was essentially this - I have a great possession for your children to inherit, and it's in Canaan.
The New is similar - I have a great possession for the children of promise to inherit, and it's in heaven
What about the new heavens and earth? Will there not be a new heavens and earth created in which that abode Christ is preparing for us is going to be co-joined?
This will be the eventual complete fulfillment to that promise of an heavenly country. I view it as separate from Heaven, to which the Saints will have access to. New Jerusalem is patterned in the earthly Jerusalem, and the inhabitants of New Jerusalem will be those of faith throughout History.
But all of these promises fulfilled are consistent throughout Redemptive History as recorded in Scripture, the only difference being how they are fulfilled according to the Age in view. In the Old Testament, the "Land" and "Rest" was a physical country. In the Millennial Kingdom the Land and Rest is a true Theocracy (which was given Israel in physical terms which she rejected, clamoring for a King of their own, and falling into idolatry). In the Eternal State it is complete, both our King and our country being that which God has promised from the beginning.
The "Church vs Israel" distinction is a false dichotomy because the Old Covenant was for an ethnic group without any regared for whether or not they were believers.
The New Covenant is for believers without any regard for ethnicity.
While I agree "The Church vs. Israel" is a false dichotomy, lol, the clear distinction between the Nation Israel (the People of God of the Old Testament in physical terms) and the Church (the People of God in this Age) is not a false dichotomy, Paul spends quite a bit of time making it clear that Israel is not the Church. This is evident in Jew and Greek/Gentile now being one man, rather than the separation we see when Israel was under the (Covenant of) Law.
This is the reason why Paul makes it a point to show that the Law did not nullify God's promises, nor do we see those under Law excluded from being deemed righteous through faith. But the Law did not intend to deem men righteous from an eternal perspective, as we see it is in Christ.
We would have to conclude that everyone under Law was excluded from the promises if take a general view which improperly imposes the possibility of receiving the promises by keeping the Covenant of Law (from an eternal perspective). Quite the opposite is true, the promises of God were maintained despite the fact that they did not keep the Covenant. Moses was under the Law, sinned against God, and was restricted from entering into the physical Promise, but, Moses was also a man of great faith and not restricted from entering into the Promised Land and Rest which was not physical (though he awaited perfection to do this).
There is no "Church versus Israel" to be found in Scripture. The distinction is, using your terminology, "Israel versus the World (Gentiles)" under the Covenant of Law, and the Church versus the World under the New Covenant.
There is, however, an Israel as opposed to the Church between the two Dispensations. Those of faith of Israel, and those of faith of the Gentiles, are made one new man, a man that did not exist prior to the Church being established on Pentecost. This is the result of the Promise of God, the eventual fulfillment of every promise given men in regards to Redemption throughout recorded Redemptive History. The promise of God in the Garden is fulfilled in Christ, Who is the Covenant relationship, the True Vine, and the foundation of the Promises of God.
God bless.