You will want to enter your home address or at least an address that you can receive mail! You should use your residential address when filling out your address on the application and you can select a separate address when filling out your mailing address.
You are unable to use PO boxes because you must have a permanent residence. The address that you submit in the application should be the same address that you will use when you file your taxes.
If your residential and billing address differ, you will be asked to enter your billing address. If your residential and billing address are the same, then yes, use that one!
Your health plan information, including options and subsidies, are based on where you consider yourself to be living the majority of the year so it's important to be as accurate as possible!