The "free C compiler" you are talking about is avr-gcc, which is the free "gcc" compiler targeted to the AVR, along with a matching libraries. The main "gcc" is written by volunteers and supporting companies, and is entirely free. The AVR port is also mainly volunteer work, but Atmel provides some support and backing for it. It is free software - you can download the source and compile it yourself. Atmel also provides packages with pre-built binaries, which is much more convenient for most users.
I don't use AVR Studio much myself (it's windows-only, and version 5 onwards are based on MS Visual Studio instead of the industry standard IDE). So hopefully someone will correct me if I get the details wrong...
AVR Studio 5 onwards come with the avr-gcc toolchain. Each release of Studio has the latest Atmel release of the toolchain. AVR Studio 4 did not come with the toolchain, but it has convenient support for using avr-gcc. In those days, Atmel did not make its own releases of avr-gcc, so the common choice (on Windows) was the "WinAVR" packaging of avr-gcc.
If you need to use AVR Studio 4 to have support for your debugger, there should be no problem using it together with the latest "command-line" avr-gcc toolchain from Atmel. I very occasionally use AVR Studio 4 myself for debugging, though I never use it as an IDE or editor.
It is also perfectly possible to use the free debugger "gdb" for debugging with the AVR. You can use the latest version of Eclipse as your IDE and debugger interface, compile with avr-gcc (using Atmel's packages if you like), and connecting to your jtag debugger using avarice. However, if you are new to this sort of thing and don't have someone nearby to help, then I'd recommend AVR Studio first.
"Atmel's AVR Studio 5 is completely free and completely free of limits - there is no code size limit for compiled code and no additional compiler purchase is necessary. The suite is completely free and has no limits."
Article has a nice animated demo of Atmel AVR Studio 5's Intelligent Editor. AVR Studio 5 also has a complete chip simulator, simulates the microcontroller core and peripherals including undocumented features. Fun to play with!