I plan arrive Beijing on 9/3, travel to Hong Kong on 9/8 and back to US from there on 9/9 or 9/10

If you have a USA passport, you can stay in Hong Kong for up to 90 days without a visa. Your 144 hours in Beijing starts at 00:01 on the day following the day of entry. You cannot take the fast train from Beijing to Hong Kong.

Thank you very much...you confirmed what I thought after reading the visa free transit guidelines some more is that you can fly US-Beijing-Hong Kong-US but not US-Hong Kong-Beijing-US. Is this correct? Thank you again for clarifying.

Either is fine if those are your only landings. But USA-Hong Kong-Beijing-Hong Kong-USA requires a visa.

Just 1 more question...after arriving in Beijing from the US and establishing visa free transit and then staying 5 days in Beijing before flying to Hong Kong for 2 days, can I then fly back to Beijing from Hong Kong for about a 3 hour layover to get a flight back to the US or do I need to leave for the US from Hong Kong?

It’s very simple. You must fly non-stop Country A - Beijing - Country B. Either A or B can be Hong Kong, but not both. It’s not difficult to understand.

Yes, you can fly from Hong Kong via Beijing to US on your return journey back home.

US - Beijing - Hong Kong - fine
Hong Kong - Beijing - Hong Kong - fine

As mentioned before, you can use a slow train on one or two of your journeys between Beijing and Hong Kong and vice versa in case that you would like to have such an experience.

Zupan, as you know, Hong Kong-Beijing-Hong Kong is not fine without a visa! I guess you mistyped...

Zupan, thank you again for your very helpful and courteous answer. I knew what you meant that on the return I can fly: Hong Kong - Beijing - US. Since I may be using some award travel for portions of the trip, my ticketless travel from US to Beijing may be one way, but I will have airline ticket and seat info available for the authorities in Beijing for my onward travel to Hong Kong. I might also use a round trip ticket from US to Beijing instead that will encompass 9/10 days but I would be still eligible for the 144hr visa free transit in Beijing as long as I don't stay beyond that window. I presume that on the return from Hong Kong my 3 hour layover in Beijing is covered by the 24 hour visa free policy.

Gerry, you are totally over-thinking this. It is really very, very, very simple. It doesn't matter if it's award travel or not. There is no such thing as "ticketless travel". If you fly Country A-Beijing-Country B then you don't need a visa for 144 hours in Beijing. Separate tickets and different carriers are fine. Hong Kong can count as Country A or Country B (but not both).

Uh, apologies... I have I mistyped. What I wanted to type was Hong Kong - Beijing - US, of course...

Everything what you have mentioned is fine and your return trip from Hong Kong to US via Beijing will definitely be covered by 24 hours visa free transit.

Award tickets are absolutely fine. It is the employee pass/buddy pass/whatever airline employees get for free (officially known as a standby ticket or nonconfirmed ticket) that doesn't count. If you fly, you will absolutely receive a ticket (even on standby), the only difference is how or whether you paid for it.