What type of personailtiy does the house have in the story "There Will Come Soft Rains"?

That's an interesting question, because the story doesn't actually have any human characters in it. The reader gets to read about a house, which is something that isn't often compared to a personality. However, the story uses anthropomorphism to give the house a wide ranging personality.  

First and foremost, I have always gotten the feeling that the house is quite nurturing. The house takes care of the cleaning, it tries to wake the family...

That's an interesting question, because the story doesn't actually have any human characters in it. The reader gets to read about a house, which is something that isn't often compared to a personality. However, the story uses anthropomorphism to give the house a wide ranging personality.  

First and foremost, I have always gotten the feeling that the house is quite nurturing. The house takes care of the cleaning, it tries to wake the family up in the morning, it cooks them breakfast, and even clears the table without complaint (despite the fact that nobody ate the meal). The house sees to the needs of each individual person with equal amounts of detail.  Even the head of the household is not ignored in favor of the children, because his card table is automatically set up for the bridge game. 

I can't fully say that the house is full of soft, nurturing qualities though.  There is a moment when the house seems to get annoyed and angry.  This occurs when the dog tracks in mud.  

The dog, once huge and fleshy, but now gone to bone and covered with sores, moved in and through the house, tracking mud. Behind it whirred angry mice, angry at having to pick up mud, angry at inconvenience.

If the house were a person, there are a few other words that I would use to describe it and its personality.  Dependable and efficient are both words that I think aptly describe the house's personality.  There just isn't a moment that the home isn't taking care of something or anticipating a future need of the now dead owners.  For example, the house takes less than 15 minutes to collect the dead dog and have it incinerated.  That's dependable and efficient.

Two o'clock, sang a voice.

Delicately sensing decay at last, the regiments of mice hummed out as softly as blown gray leaves in an electrical wind.

Two-fifteen.

The dog was gone.

In the cellar, the incinerator glowed suddenly and a whirl of sparks leaped up the chimney.