Why has it been so hard for China to build aircraft jet engines?

 Empirical evidence proved it is more difficult to design and build aircraft engines than it is to build atomic bombs or send satellites to space. The jet engine is HARD.

It took many decades for US, British and French engine makers to perfect the technologies that they use today. China couldn't just acquire the plans and start making them. Creating turbine parts that can survive extreme heat has been a major engineering challenge. Meeting it has required fundamentally rethinking the material structure of the turbine blades, making metals do things that they do not normally do in nature. Turbine blades may be operating in temperatures far exceeding their melting point, and thus must be cooled to typically 85% of melting temperature to maintain integrity. Blades must be cast with intricate internal passages and surface hole patterns needed to channel and direct cooling air within and over their exterior surfaces.

The turbine blades in a modern jet are a single metal crystal. Even if you have a jet and can take it apart that tells you nothing about how to cast the turbine blades, and you can have a turbine blade that looks fine but self destructs after 200 hours. As an example, the load on a single turbine blade that's only 3 inches long can be as much as 35 TONS due to the centrifugal force on a turbine spinning at 20,000 rpm at temperatures over 1,100F. Only single crystal nickle-steel can sustain that kind of load for 2,000 hours without fail.

One of the biggest reasons why China is so lagging behind in jet engine is the Culture Revolution, which thousands of scientist/engineers were purged, thus created a talent gap which China took a generation to pick up. By the time China got its sanity back, it's already in the 1970s. Chinese engines have to compete with mature jets already in the market place, and US and UK engines didn't. This let them make profit while they were making junky engines. China doesn't have that luxury.

China has been making jet engines for decades, starting with WP-5 in 1956, which is a licensed copy of Soviet VK-1, which itself a knock-off from Rolls Royce Nene. China and UK signed a license deal in 1974 for the Rolls Royce Spey engine, but since UK committed to transfer technology, but not manufacturing know-how, by the time China overcame all the technical difficulties associated with manufacturing the jet engine, it was already mid-2003. China probably will surpass France by end of the 2020s, as China has multiple engine developments in the pipeline.

Can small private jets fly internationally?

 Yes, they are treated the same way as most general aviation aircraft, and yep, they do it all the time. International flight can be as short as from Buffalo, NY, across the lake to Toronto, Ontario, Canada, or from Arizona to Mexico, and a Cessna Citation X can fly from Detroit, Michigan to Toronto, Canada, or from Phoenix AZ to Mexico City. It does not even need to be a jet. Even you can fly internationally in a Cessna 172.

The smallest business jets are the Very Lights (VLJs) like the Hondajet and Phenom 100, and those aircraft have a max range of 1,500 miles (ish). I’m assuming by international you mean long trips across oceans, and the small jets won’t have life rafts or some cockpit equipment required to fly across the ocean, which is fine because their cabin about the size of a train compartment. Also there’s limited luggage room, potentially filled largely with survival equipment required for 100nm over water flights. From the US to Europe, we’d preferably go Bangor, Maine to Goose Bay to Keflavik, and from there it depends on final destination. In general, trips like this make sense only for ferrying or special purposes.

A private jet can fly around the world with careful planning, but due to the range of most small jets, multiple stops for refueling must be made and the route would have to be carefully planned not only for fuel and range but also to ensure that one is flying within the airspace of friendly countries. It totally depends on the capability of the airplane and the length of the route you are asking about, and whether they are permitted will depend again on the abilities of the airplane, the crew qualifications and on‑board equipment, and will they be permitted to land at the International airport.

In Canada, any general aviation aircraft with fewer than 15 persons aboard and no cargo or commercial goods is cleared through customs and immigration in the same manner. General aviation aircraft are cleared at their fixed base operator, and some airports have a dedicated customs ramp to clear private aircraft. Between 2 and 48 hours prior to arrival, the pilot in command will call the Telephone Reporting Centre with aircraft make, model and registration, last airport of departure, airport of arrival, estimated arrival time, passenger and crew manifest, and declarations. If the aircraft is released, everyone is free to disembark, and if the aircraft is referred for inspection, two or more officers will be dispatched to attend the plane.