Yes - they are both part of Star Alliance and have reciprocal arrangements
Yes, these two airlines are part of the Star Alliance.
Singapore is a member of Star Alliance. Its U.S. members include United Airlines and U.S. Airways. U.S. Airways may not remain a member, since it is merging with American Airlines, but United will. Members of any Star Alliance frequent flyer program can earn miles for flights on any member's flights and can redeem miles/points for flights with any member.
Yes - they are both part of Star Alliance and have reciprocal arrangements
China Airline's frequent flyer program is called Dynasty Flyer. Partner airlines are Mandarin Airlines and China Southern Airlines.
American Airways offers a frequent flyer program that can be used on any of their flights, including international.
American Airlines Advantage, Delta Sky Miles, and US Airways Dividend Miles are rated as very good programs by Business Travellogue. The tell you what the rewards are and how to get them at http://www.businesstravellogue.com/best-frequent-flyer-programs.
Apply on the airlines website
The discount one will receive from being a frequent flyer aboard US Airways will depend on the number of miles one has undertaken. Usually, however one will use their frequent flyer points towards free flights.
United Airlines is an airline company. It recently merged with Continental airlines and uses their logo, computer reservation system and Frequent Flyer numbers. It retains the airline name United Airlines.
The vast majority of the major airlines around the world have frequent flier programs.
The easiest way is to apply on-line at the airlines website.
With careful planning, this trip can be done for as little as 20,000 United Airlines FF miles.
Typically this would be 100,000 miles round trip.