The MPI_Init documentation of MPICH is giving some hints:
The MPI standard does not say what a program can do before an MPI_INIT or after an MPI_FINALIZE. In the MPICH implementation, you should do as little as possible. In particular, avoid anything that changes the external state of the program, such as opening files, reading standard input or writing to standard output.
BTW, I would not expect MPI_Init to do communications. These would happen later.
And the mpich/init.c implementation is free software; you can study its source code and understand that it is initializing some timers, some threads, etc... (and that should indeed happen really early).
To me it seems perfectly reasonable for processes to do a significant amount of calculations before starting the communication with each other.
Of course, but these should happen after MPI_Init (but before some MPI_Send etc).
On some supercomputers, MPI might use dedicated hardware (like InfiniBand, Fibre Channel, etc...) and there might be some hardware or operating system reasons to initialize it very early. So it makes sense to call MPI_Init very early. BTW, it is also given the pointers to main arguments and I guess that it would modify them before further processing by your main. Then the call to MPI_Init is probably the first statement of your main.