Any airline in the Skyteam alliance - but particularly with AirFrance and KLM
Delta offers frequent flyer miles that they call Delta SkyMiles. You can find out the benefits, rules, and answers to FAQ's at http://www.delta.com/skymiles/index.jsp
China Airline's frequent flyer program is called Dynasty Flyer. Partner airlines are Mandarin Airlines and China Southern Airlines.
There are a number of frequent flyer programs available from all major airlines. I personally like the Hawaiian Air Miles program as well as the Delta Sky Mile program.
Yes, AMEX are heavily involved with frequent flier programs. They have a very strong agreement with Delta airlines.
Reviews covering Delta Airlines' frequent fly program, known as SkyMiles, can be found on websites such as The Points Guy, About, and The Frequent Flyer Authority. Generally, reviews of the program are negative, citing a poor rate of miles earned compared to other companies. Delta revamped the program in 2013 in an attempt to make it better.
Apply on the airlines website
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.
Yes, they are both part of the Skyteam Alliance and the miles may be used inter-changeably.
No you can not. United Airlines is part of Star Alliance and Delta is part of Skyteam. They are competitors.
The vast majority of the major airlines around the world have frequent flier programs.
You can apply for most frequent flyer programs on the airline's web site. Be sure to present your frequent flyer card when you check in for each flight, because sometimes it can be a hassle to get credit for miles retroactively.
The best frequent flyer credit card is the Gold Delta SkyMiles from American Express. You can apply for the Gold Delta SkyMiles Card online from the American Express website.