|All of your questions really depend on what you are trying to accomplish when programming in Objective-C. If you are just using it as a language the answer is yes you can program with it on windows and linux.
However if you are looking to develop with the Cocoa frameworks and/or for iOS platforms then you will need:
1 - Apple hardware (mini, macbook, iMac, Pro)
2 - XCode will be the IDE that you most likely will want to start with if you are new and learning.
3 - You can get XCode from the App Store on your OSX machine.
Hope this answers your questions.