A list of all games created by EA (Electronic Arts) is available on the EA website, or a more comprehensive list is available to view on Wikipedia. There are also several other websites with lists of all EA games created, but these may or may not be incomplete.
Games owned by the publisher "Electronic Arts" typically use EA servers. Some games that use EA servers include Sim City, Mass Effect and Battlefield 3. Electronic Arts also has it's own digital distribution service known as Origin. Origin games use EA servers.
No. When you make an EA account you will then be able to manage all of your games on different platforms through the EA Origin. Everything is free but it does cost money to download or buy games unless stated otherwise.