Hi rebelizdav!
There's a great deal of information to learn about this aspect of the game. First, I don't know why your first house showed up in your second game. If you started a whole new game from the beginning, it would be a separate game file and should have nothing but the original town. However, I've never had my one and only Sim die and end the game. What happened when she died? Did you get an option to create a new Sim from that point? Or was the game just finished and done?
When you start a brand new game and create a new Sim, a copy of that Sim goes to the library and is available for you to play again if you choose. Or you can place your previously created Sim(s) somewhere else in town and let her be around without playing her. This is done through Edit Town once you start a game.
You can save a copy of your house through Edit Town, also. Once you're in Edit Town, click on the lot. Then click on the Save to Library button. This will let you save the household with the house to the library and, again, it will be available to you in another game.
There's probably a way to save just the house, but I'm not clear on how to do it, so I'll let someone address that part with you.
I hope this is helpful and not too confusing.