LAX-SHanghai (3 days) - Manila - LAX on Philippines Airlines all the way. Is this eligible ?

I think, that your Philippine Airlines flight between Los Angeles and Shanghai is not direct, but has a transit (transfer) in Manila. If I'm not wrong, that would not be a transit via Shanghai, but a return flight on the route Philippines - Shanghai - Philippines looking by the book the law regulating 144 and 72 hours travel without visa. I know it could appear to you as something weird, but the Chinese authorities would definitely deny entry in China without a pre arranged visa.

In case that I'm wrong and your Philippine Airlines flight is a non stop one from Los Angeles to Shanghai, then you are perfectly within the legal frames of 144 hours free travel.

LAX layover in Manila continuing to Shanghai (3 days) then onwards to Manila for 1 week, back to LAX

I was afraid of this situation. In short, if you have already booked the tickets, you have to obtain Chinese pre-arranged visa because the travel plan in this form does not qualify for 144 hours visa free travel. The key problem is in the fact that you will arrive in Shanghai from the Philippines and you will return there after 3 days stay. This is not allowed although you will just have a layover in Manila.

Now that I’m in Manila, Will a fly to Shanghai then Bangkok then Manila qualify for the 144 hours hour free travel?

Definitely, yes. This is the situation that is perfectly within the legal frame for 144 hours free travel.

Manila (Philippines country A) - Shanghai (China country B) - Bangkok (Thailand country C). This is the right formula to obtain this convenience.

By the way, your flights to and from Shanghai have to be nonstop!

Of course. Option Manila - Shanghai - Bangkok is definitely within the legal frame for 144 hours free travel. Anyway, you should have in mind that the flights to and from Shanghai have to be nonstop!

Can I use any carrier, like PAL which has a direct flight to Shanghai, also a direct flight to Bangkok? Or am I limited to use Chinese Airline?

PAL does not fly Shanghai to Bangkok non-stop. But any carriers are fine.