Confirming route and repeat use

We have several problems here... As far as I know, at least at the time being, there are no direct (nonstop) flights between US and Chengdu. Even if there are, you are not eligible for 144 hour visa free transit because your next destination is Tibet region that is an integral part of the mainland China.

It is important to know that the main condition for 144 hour visa free transit eligibility is that your immediate points, before and after your Chinese destination are in two different countries or eligible territories (eligible territories are Hong Kong, Taiwan and Macau). For example, if there were direct (nonstop) flights between Chengdu and US, then your itinerary mentioned under 4 and 5:

Kathamndu (Nepal) - Chengdu (China) - US

Would definitely be eligible for 144 hour visa free transit.

So, you have to obtain a Chinese visa in advance because the Chinese visa obtained in advance is a condition that you can obtain Tibet Travel Permit.

Thanks for your answer.
Right now there are flights to Chengdu via Seoul.
If we transfer in Seoul to a direct flight to Chengdu, then go to Nepal, that would qualify, correct?

Then we could get a visa from Nepal into Tibet.

But we would need a normal China visa to return to Chengdu to fly out?

Thank you.

Yes, you are eligible for 144-hour visa-free transit via this route. However, if you still want to go to Tibet, a China visa and Tibet travel permit are must. By the way, applying a China visa in Nepal is a little hard as a tourist.

As already mentioned by Gloria, if you intend to fly:

Seoul - Chengdu (stay up to 6 days) - Kathmandu (stay up to 6 days)

and

Kathmandu - Chengdu (stay up to 6 days) - Seoul

You do not need any kind of visa, but you will be covered by 144 hour visa free transit scheme. That is for certain.

I have intentionally written Seoul because in 144 hour visa free transit scheme only the mentioning of immediate points before and after your Chinese destination matter and they have to be in two different countries or eligible territories (eligible territoris are Hong Kong, Taiwan and Macau).

So, if you skip your visit to Tibet and retain just Chengdu as a transit point in China, you do not need any kind of visa.