Right, like Metro said, you have to call the babysitter each time you need one. However, they don't stay 24 hours if your Sim is out of the house. They stay only until your Sim returns home. You can also call a babysitter to come stay with the baby just so your Sim can get some rest or relax with skills. I think this is where Metro is thinking of the 24 hours. In this situation, the babysitter will stay at the house with you until around midnight, I believe.
When your Sim gets married, you are given the option to choose which house to live in. If there are others living in his house at the time, you can move them to your old house if there are enough beds. The only time you won't have a choice of houses is if you marry a Sim who doesn't have a house, like one of the NPC Sims (maid, fireman, etc.).