How can I set up registration and payment for an ongoing event that happens every saturday
This question has a lot of aspects. I will focus on the fact that the fee is based on the household. Does it mean that the household is your actual customer and that you are not interested in the details of the participant? And you only want to prevent that the event is overbooked? If this is the case you can use Price Fields. 




Create as part of a Price Set a Price Field set a Text/Numeric Quantity field (in my example Seats).
Enter the participant count, make this 1 (see (1)).
Set the price on 0.00 (this field is only used to track participants see (2)).
Uncheck the Display Amount option.


Now in the online registration form you can enter a number in the Seats field and this will add up to the total number of participants.