A "house" is essentially a container -- for a person, a family, several families or even a large group with shared interests (think of the term "houses of worship.") In the Bible, a "house" is also a container, but for spiritual things rather than natural things. In various uses a "house" can represent part of the mind, the whole mind, a whole person or even a church. The other nuance to the word "house" is that it is generally used in regards to our affections and desires rather than our thoughts and principles. This makes sense; we tend to engage our thoughts and rationality when we are out in the world doing our work, but when we are inside our houses we are driven most by love for our families and the desire to be good to those we love. So "house" tends to represent the things we want and care about -- which are ultimately the things that define us.

2003. 'As for Me, behold, My covenant is with you' means union of the Divine Essence with the Human Essence. This is clear from the meaning of 'a covenant' as conjunction, dealt with in 665, 666, 1023, 1038. Here it is the union of the Divine Essence with the Human Essence, as is evident from this meaning and from the internal sense of what has gone before, and so from the actual words 'My covenant is with you'.