I'll attempt to answer this since it's the shortest.:-)
Jacob was of course renamed Israel:
Gen 32:28 And he said, Thy name shall be called no more Jacob, but Israel: for as a prince hast thou power with God and with men, and hast prevailed.
Jacob of course had 12 sons whose offspring are called collectively "Israel"...or the house of Israel.
So this is just a way of referring to Jacob's (Israel's) offspring as opposed to the kingdom of Israel eventually became a political entity separate from the Kingdom of Judah.
In other words, all tribes are part of the house of Israel. But not all tribes were part of the Kingdom of Israel.
This was true after Solomon....but under Saul, David and Solomon all the tribes were part of the Kingdom of Israel.
Douglas is correct in that by saying, "House of Israel"... it's like saying "All of Jacobs descendants". There is also a house of Abraham....which includes Jacob (Israel), Esau, Isaac and Ishmael.....and others. [Genesis 25:1] On the other hand.....the House of Isaac would only include Jacob and Esau.