This is an example of what is readily available via a California SecState online entity search:
Entity Name: GOOGLE INC.
Entity Number: C2474131
Date Filed: 11/07/2002
Entity Address: 1600 AMPHITHEATRE PARKWAY
Entity City, State, Zip: MOUNTAIN VIEW CA 94043
Agent for Service of Process: CORPORATION SERVICE COMPANY WHICH WILL
DO BUSINESS IN CALIFORNIA AS CSC - LAWYERS INCORPORATING SERVICE
Agent Address: 2710 GATEWAY OAKS DR STE 150N
Agent City, State, Zip: SACRAMENTO CA 95833
Other data you included in your filing is also publicly available but not searchable online as of this posting. CA SecState does provide this data of all historical company records on disc for about ~$125-$250 to anyone who wishes to buy that data.
Could I use a service like the UPS Store to get a mailbox and use this
address for everything, for example?
would I need to fill in my home address as the location of the LLC's headquarters or is it > correct to list my registered agent's address in Delaware?
You should ask the SecState's office about this. They might reject the filing because the address looks like it's a mailbox, wasting your time and filing fee.
That said I've used the agent address as the principal address in California with no problems (I operate from there anyhow, but it's not like they would know if I didn't). To my knowledge, there's no penalty expressed in the California corporations code for listing your agent address as your principal address, nor any penalty expressed in the documents provided by the SecState's office. Make sure you can receive service of process if you're sued and respond to tax notices using the agent address you put down.
Ultimately, the spirit behind asking for the actual entity address is to be able to locate the company in case of a lawsuit and to enforce unpaid taxes. Someone suing you could claim, however, that the entity you declared as shielding you from liability is not valid because it is not registered with the same principal address located where you are actually doing business. I don't know how well this argument stands in court (imo it seems pretty weak), but that's the only issue I would be concerned about. Does this concern outweigh your privacy concern?
Disclaimer: This is a personal opinion and should not be taken as legal advice.