I store the bottles/teats in a big box after using them and washing them. And yes, just before I use them, I sterilize them.
Many mamas have different methods. But I think sterilizing them just before use is more hygienic. In fact, according to Avent's sterilizer instruction manual, they only guarantee the bottles to be clean for the 3 hours just after sterilization.
So, it can be more trouble, e.g., if your BB needs to be fed for 6 times a day, then you need to turn on the sterilizer 6 times a day. (Usually for me, I would do so one hour before each feed). But after I've got used to it, it's okay.