If Airbus Beluga can fly with two engines, why doesn't Boeing 747 and Airbus A380 fly with two engines?

 It is the difference between weight and volume. The Beluga can carry huge components, but they are not very heavy. The Beliga is designed to carry oddly shaped, and therefore hard to carry components, specifically, wings and fuselage sections. The thing about the Beliguas is they exist to move big and awkwardly shaped components around Europe, particularly ferrying wings from the UK to France.

There's much more empty space inside a Beluga than there would be on the other two aircraft types. The Boeing 747 was designed in a time when engines were weaker than they are today. This airframe was designed back in the 1960s. Back then engine reliability was at nothing like the exceptionally high level we are used to in the present day. Both the B-747 and A-380 as well as the A-340 have 4 engines, not so much for the extra power, but in their early design days, you could not fly over oceans with 2 engines for more than about 60 minutes. When the 747 was first designed and built, engines in the 100,000 pound thrust capacity didn't exist, nor was anything even close to that available.

The Airbus A380 on the other hand, does need 4 engines. The A380 needs 280,000lbs of thrust. So as a twin-jet it would require two 140,000lb thrust engines. Even today there is no engine either in service or proposed that would deliver that much power. Due to its weight it needs powerful engines, but also because it's lower to the ground, the engines needed to be smaller.

The range and the MTOW needed is not as high as an A380 or B747 require, therefore two engines are fine for the Beluga. You cannot swap out 2 engines onto a 4 engine plane without re-engineering virtually every part of it. This would be too costly to even examine.

Why is the Boeing 777 so successful?

 The 777 was a big hit as the first twin-engine jet with the capacity to challenge the 3 and 4-engine jets of the time. The Boeing 777 was the first twin-engine plane to carry over 300 passengers in a typical configuration. Twin-engine planes have less mechanical parts and are more fuel efficient. Boeing had a much different vision than Airbus, for they envisioned a two engine jet that would be highly efficient and could carry upwards of 300 passengers.

The reason that so many airlines wanted the 777 was because it could travel extremely long distances, even longer than the 747, and it could carry almost as many passengers in the -300ER variant, while being extremely fuel-efficient. The 777 also had a very long ETOPS rating, which was eventually raised by the FAA only a few years after the launch of the aircraft. The Boeing 777 offers the best of both worlds - long range and good payload capabilities. Because with two engines, it can carry over 300 passengers upwards of 8,000 nm. It is therefore better matched with the route capacity desired by most airlines, and its direct operating cost per available seat mile is Less than that of the A380 and the 747–8.

The passenger version of the 747–400 was discontinued in 2005, the same year that the 777–300ER entered passenger service, because both planes have the same capacity and range. The Airbus 350–1000 is comparable to the B777 but Boeing 777 hit the market 20 years earlier than the A35X. It accumulated over 20 years of flight experience proved to be a reliable jet and versatile enough to go to any international airport.

During the design phase of the 777 project, representatives from the customers were directly involved in the decision making. The 777 project involved one of the earliest uses of computer aided design. So confident were they in the efficacy of collaborative design by computer, that they skipped entirely the mock-up phase. To date there have been more 777s built than 747s and the profit margins on the 777 were apparently enormous.