You say application? I'm going to asume you mean just games rather than any old application.
For games in general (excluding iOS for the time being) C++ has been the staple for a long time in the games industry, and as such there are many many tools available for it, physics, graphics etc. C# is looking good atm, and the XNA platform is easy to use, You can also use C# with Unity3D if you are looking for a pre-built solution.
I'm not sure how many games on the iOS use Obj-C, I wouldn't be surprised if alot of them are C++, All the ones I know for certain are in C++ or Unity and I dont know anybody who's made a Obj-C game for the iOS (Which is more todo with the circles I'm in than anything else). The game I've been working on for iOS is all C++.
Obj-C and C++ talk to each other quite happily the xcode compiler is smart enough to know what is what.
Xcode is almost a must. The only exception I can think off at the top of my head is if you use Unity, or Corona (which you might want to look into, I've heard flash devs talk nicely about it)
Obj-C You will need to know a little at the very least (this can just be cut and pasting)
C++ Has a big following, however you will need a little Obj-C to get your game working even if you use C++.