A good friend asked me for my thoughts on the above subjects...
Dispensationalism does recognize some important biblical theological markers and shifts (i.e. dispensations). That said, I disagree with them on their understanding of the OT and thus the way they would view Israel and the Church.
In a nutshell, whereas dispensationalism sees Israel and the Church in sharp discontinuity, I see them in perfect continuity, but only after I alter some definitions. They view Israel as always corporate or national Israel. But Israel is only the "true Israel" spoken of by Paul in Romans 9. National Israel has never been true Israel. When Paul says, "not all Israel is Israel," or, "not all national Israel is truly Israel, in the sense that Israel is God's chosen people," he is saying, "true Israel is a part of national Israel." We can go further. Since he uses examples that cover the entire Old Testament history of Israel we can say that Israel has never been truly and completely the people of God, or true Israel.
Look closer. This is more true than you may immediately think. Israel NEVER believed God or received his prophets or His words. NEVER. From Abraham to Moses to the Exodus to the Conquest to the Judges, the Kings, the Exiles, the Prophets, the silent years, the Christ, Israel as a people continually rejected God and His word from being over them, even in the midst of unprecedented acts of God. Israel IS NOT Israel. Jews are not the people of God, but rather, again with Paul, "The remnant shall be saved."
Promises to "Israel" are to the God-fearing remnant inside of national Israel. Again, this makes sense upon immediate reflection. Only the God-fearing remnant was obedient and could receive the promises! Israel will never receive the promises because the nation is not true Israel.
To go even further, not only is Israel not true Israel, but Jesus is true Israel. He is the seed of Abraham, the seed of the Woman, the seed of David and Solomon. He is the suffering servant. HE IS TRUE ISRAEL, so then we are true Israel, in Him.
Prophetically speaking then, all the promises to true Israel in the OT have been or will be fulfilled in or by true Israel, Jesus, including the church.
These prophecies have been or will be fulfilled in a manner consistent with the original author's intent, which even a cursory reading can see is probably not "literal." In fact, Dispensationalists use contradictory criteria when they attempt to read apocalyptic literature both "normally" and "literally." In this case, at least, it is one or the other.
With Christ at the center, the Old Testament stops being about the nation of Israel and starts being the story of God's people, true Jews, within the nation, who looked for and longed after the Christ, whom we learn is Jesus.