Just so everyone is clear - there are no restrictions on household additions and moving in this event. However, the achievements you count is the list presented in Options at the end when your initial Sim turns Adult. There is no carrying over of achievements obtained from prior households.
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EDIT: Okay, someone reminded me that achievements are tracked for each save file, so what I said above is misleading. Bottom line: you can pretty much do whatever you want to do to rack up achievement points aside from opening up the cheats panel
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EDIT 2: Since it's been at least an hour since I have squawked at everyone I needed to also remind you that some achievements will be impossible given your original CAS Sim is your timekeeper. In other words, you cannot leave him/her behind in someone's home. You have to keep them because it's when they age up that you stop playing.