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Also, many from the Northern kingdom moved south into Judah when the northern kingdom turned to idolatry.

Yes, they did, that would be why there are references to other tribes in the predominately Judah remnant during the New Testament. But the Northern house was still population heavy compared the Judah.

I suppose the "house of Israel" could be used collectively, after all, they all were Israel, and Matthew 15:24 would tend to support that.

"But he answered and said, I am not sent but unto the lost sheep of the house of Israel."

But by the same token He would have included the greatest mass of Israel, for they were indeed lost, and there are indications that one or more of the disciples in addition to Paul were up in Europe. Jesus was sent, as He said, not to just to Judah and Benjamin, but all of Israel. I'd doubt that the location of the rest was hidden from Jesus.

There is also evidence of remnants of the remaining house of Israel in Europe and Assyrian records that indicate that direction for the remnant after Assyria was defeated. They would have had three/five hundred years to get there.

I don't think the chapters of any of the four Gospels are in chronological order, so it's probable that when Jesus instructed the apostles and charged them to go teach, it was referring to their life mission, not just a trip out to the towns in the region.

***I suppose the "house of Israel" could be used collectively, after all, they all were Israel, and Matthew 15:24 would tend to support that.***

Here are some better references.

2Ch 20:35 And after this did Jehoshaphat king of Judah join himself with Ahaziah king of Israel, who did very wickedly:

2Ch 21:2 And he had brethren the sons of Jehoshaphat, Azariah, and Jehiel, and Zechariah, and Azariah, and Michael, and Shephatiah: all these [were] the sons of Jehoshaphat king of Israel.

Was Jehoshaphat BOTH king of Judah and Israel? These say he was.

2Ch 28:1 Ahaz [was] twenty years old when he began to reign, and he reigned sixteen years in Jerusalem: but he did not [that which was] right in the sight of the LORD, like David his father:

Ahaz, in Jerusalem was king of Judah.
But he was ALSO King of Israel.
2Ch 28:19 For the LORD brought Judah low because of Ahaz king of Israel; for he made Judah naked, and transgressed sore against the LORD.

Jhn 3:1 There was a man of the Pharisees, named Nicodemus, a ruler of the Jews:
Or was he!
Jhn 3:10 Jesus answered and said unto him, Art thou a master of Israel, and knowest not these things?

And Paul..Israelite or Jew?

Rom 11:1 I say then, Hath God cast away his people? God forbid. For I also am an Israelite, of the seed of Abraham, [of] the tribe of Benjamin.

2Cr 11:22 Are they Hebrews? so [am] I. Are they Israelites? so [am] I. Are they the seed of Abraham? so [am] I.

Act 21:39 But Paul said, I am a man [which am] a Jew of Tarsus, [a city] in Cilicia, a citizen of no mean city: and, I beseech thee, suffer me to speak unto the people.

Act 22:3 I am verily a man [which am] a Jew, born in Tarsus, [a city] in Cilicia, yet brought up in this city at the feet of Gamaliel, [and] taught according to the perfect manner of the law of the fathers, and was zealous toward God, as ye all are this day.

Upon return from captivity Ezra 6:17 says they offered dedication sacrifices for ALL ISRAEL and according to the number of the tribes of Israel.

I think that is plenty to show that enough returned of the whole house of Israel returned to Judah to be considered Israel.