yes your right on the F visa.
you can use an invitation letter to apply F visa there in states.
but i think you already known, F visa only allow you to stay at most 1 year straight. and after that you have to leave the country once. then come back extend it.
(ofcos doesnt matter where you go, we just need the stampt at the boarder)

i suggust you to use US passport, cos the government charge is 940 RMB for any entry times for US citizen.
If you want multiple entry, obviouslt US passport would be cheaper.
But if you want single entry, UK passport is cheaper.

sorry i dont know anything about how to register dogs...

and i can help on the invitation letter and further extension.
you can contact me for all information. =)

After you move to China, will you travel often out of China mainland? Or you prefer a continuous stay in China mainland?

For your 10-day short visit, you can keep using your multiple-entry L visa.
BTW, when will your L visa expire? If it is a quite satisfying visa which was issued not long ago, you can keep using it for the trip after next one, too.

Or during your following stay in China, you can get an F visa notification letter which takes 5 working days (7 calender days). It is better that you do it right after you enter into China mainland. With the original letter issued by the authority, you can apply for an F visa in the US.
Please double check with the Chinese embassy/ consulate in the US whether they don't care that a US citizen using a UK passport to apply for a visa in the US or not.

Whatever the visa type the foreigners have, after they come to China, they need to register at the local police station and get a registration form of temporary residence.
Thus don't worry about

As for your puppies, it is not hard to register when they safely come to your apartment in China. The real problem is: they will suffer from both the long travel and the long quarantine at the customs (It is said the quarantine lasts 30 days in Beijing. It is different from city to city, though. Please double check it as I don't have personal experience in dealing with it).

Flatly speaking, without company- or organization-sponsored employment in China on your own merits (enter on a Z visa), OR being enrolled as a long-term student (enter on an X visa), you will NOT BE ELIGIBLE FOR A RESIDENCE PERMIT. Full stop. Chinese law does not recognize freelance or "self-employed" foreigners, unless you want to jump through hoops and pony up cash to form a local company and then be able to employ yourself. However, it doesn't sound like in your situation you really need to have "residency". The dogs can officially come in under your partner's paperwork and work visa. Don't think of residency in China in the same sense it's used in the USA--it ain't that way at all. Even people on Residence Permits have to do the new visa shuffle every year, it's just that their organizations pay for it and help juggle the paperwork process. To the Chinese, we are all just passing through--the ones on the Residence Permits get to pay Chinese taxes, though.

Your best bet is to probably apply in the USA for a 12-month, multiple-entry F visa, with a duration of stay of at least 90 days (easy to get, I've got one of these issued from the Houston Consulate via an agent there). You may be able to get longer duration of stay per entry--say 120 days. You'll have to do this on your US passport. You will have to periodically leave China to reactivate another entry, but I haven't found this to be a hassle at all, as I generally holiday or have other interests outside China periodically. In a pinch, you can round-trip from Beijing to Mongolian border in 36 hours and about RMB 500 total (US passports don't need Mongo visa).

Also, with judicious planning, the terms of the typical F visas issued in the USA don't really expire on 12 months--you can actually get as much as 15-16 months out of one of these, as your last ENTRY (not exit) must be before 12 months is up. You'll need a letter of invitation for the F application, which can be concocted for you by a friendly Chinese or an agent here can do (for a price), during your upcoming short visit. Get one and take it back with you.

Your backup option is to get a single entry F visa on your US passport, which can be done when you are back in the USA--try to get a 60- or 90-day duration to buy time. Can also be done quickly in HKG on your US or British passport. Then once in China, use a gray-market visa agent here to convert it to a new F visa which can be procured for as long as 12 months, no limit on duration of stay, and is multiple entry. Check other recent posts for prices--it won't be cheap. These are your best 2 options, assuming you don't want to work regular employment and you don't want to be a full-time student.

Unless you are legally employed, a matriculated student or the spouse of one of the former or a Chinese national, it is not possible for you to get a residence permit. Long-term F visas are not difficult to come by.