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1: are there any restrictions on the 72 hour stop over on the length of time that must pass between visits?
i.e. theoretically, could one fly to Japan via Beijing, stay in Japan just a few hours, then fly home via 72 hours in Beijing again?

My more serious thinking here is of going to Japan with a stop over in one Chinese city and then 2 weeks later as I head home doing the same in another city.

2: Does your transfer have to be with the same airline?
i.e. could I book a flight purely to Beijing and back...with budget airline flights to Japan and back being what keeps me from staying in China for 2 weeks.

A partial answer.

In 2014, we flew London - Paris - Beijing with Air France, spent a few days in Beijing, and flew on to Manila with Philippine Airlines. The tickets were not just with different airlines but bought separately. We had no visa problems.

A few weeks later, we went home on the reverse trip and again stopped in Beijing.

This answer is not quite perfect for you since we had single entry visas which we used on the way out so that we could stay more than 72 hours. For the return trip, we did not have a visa and relied on the stop over visa.

One thing to note on the stop over visa is that it is only valid for Beijing. So, for example, you should not go to visit the Great Wall since that is outside Beijing.

We did not stay there 2 weeks and I don't see how you could with your plan. You will have two visits with a maximum of 72 hours with a gap between them. I guess that you could try going in and out multiple times but you may have a problem. This sort of visa dodging is disliked by many countries and, I think, the 72 hour visa does not allow you return to where you came from.

Why not get a visa? It is a bit slow and expensive but I would expect still less hassle and cheaper than trying to get around it with complex travel plans.

1: are there any restrictions on the 72 hour stop over on the length of time that must pass between visits? i.e. theoretically, could one fly to Japan via Beijing, stay in Japan just a few hours, then fly home via 72 hours in Beijing again?

There doesn't seem to be any formal minimum period. But I would imagine that repeated transits of 72 hours within a week might raise questions. If you need to do this, I think it would be better to obtain a proper visa.

My more serious thinking here is of going to Japan with a stop over in one Chinese city and then 2 weeks later as I head home doing the same in another city.

I have done this a couple of times; it doesn't seem to cause any problems.

2: Does your transfer have to be with the same airline?

No.

i.e. could I book a flight purely to Beijing and back...with budget airline flights to Japan and back

Yes, I have done this (well, not a budget airline, but across separate tickets).

The point of this scheme is to enable transiting visitors to enter China for short, infrequent visits. If you need to spend a lot of time in China, it is probably better to purchase the appropriate visa.