Are promises to the nation of Israel applicable to Christians?

In general, no promise to Israel is assigned to Gentiles, except in an in-direct sense. Israel is promised a Messiah, and Gentile believers share in that promise. Israel is promised a kingdom and Gentile believers share in it as well. Apart from these general truths, we must be reticent to assign promises spoken to Israel to Gentile believers. As Paul said:

Rom. 9:3 For I could wish that I myself were accursed, separated from Christ for the sake of my brethren, my kinsmen according to the flesh,
Rom. 9:4 who are Israelites, to whom belongs the adoption as sons, and the glory and the covenants and the giving of the Law and the temple service and the promises,
Rom. 9:5 whose are the fathers, and from whom is the Christ according to the flesh, who is over all, God blessed forever. Amen.

All promises are given to Israel, and Gentile believers are blessed to be grafted into those promises and receive the “scraps from the table” as Jesus acknowledged:

Matt. 15:22 And a Canaanite woman from that region came out and began to cry out, saying, “Have mercy on me, Lord, Son of David; my daughter is cruelly demon-possessed.”
Matt. 15:23 But He did not answer her a word. And His disciples came and implored Him, saying, “Send her away, because she keeps shouting at us.”
Matt. 15:24 But He answered and said, “I was sent only to the lost sheep of the house of Israel.”
Matt. 15:25 But she came and began to bow down before Him, saying, “Lord, help me!”
Matt. 15:26 And He answered and said, “It is not good to take the children’s bread and throw it to the dogs.”
Matt. 15:27 But she said, “Yes, Lord; but even the dogs feed on the crumbs which fall from their masters’ table.”
Matt. 15:28 Then Jesus said to her, “O woman, your faith is great; it shall be done for you as you wish.” And her daughter was healed at once.

It is Israel’s “table” and we are granted the blessing to receive the crumbs, but what glorious crumbs they are. As Paul adds in Romans 11:

Rom. 11:12 Now if their transgression is riches for the world and their failure is riches for the Gentiles, how much more will their fulfillment be!

Scripture quotations taken from the (NASB®) New American Standard Bible®, Copyright © 1995, 2020 by The Lockman Foundation. Used by permission. All rights reserved. www.lockman.org