After compiling the C program, I can take the binary executable and run it on another system where gcc is not loaded, for example, the Ubuntu field?
Partly true. Yes, you do not need the compiler itself, but you need the C runtime library. In any case, both the C compiler and the C environment are part of POSIX, so you cannot find a Unix system without them.
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Depends on the platforms and libraries installed on the platform. A compiled executable will most likely be dynamically related dependencies between libraries, for example, libcand what was associated with it at compile time.
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What platforms / compilers are involved and where do you want to move this binary?