first "The Switch from Real Mode" so we are in real mode here so why do we use ecx and others 32 bits elements ?
Because 32-bit operations are convenient, even in real mode.
Other question, at "Entering the 64-bit Submode" what is the difference between IA32e mode and 64-bit long mode ?
The wiki page was wrong, it's supposed to say "compatibility mode" instead of "IA-32e". (IA-32e is Intel's name for AMD64.) I've corrected the wiki page.
I understand the first allows us to execute 32bits code but does it allows us to execute 64bits code also ?
You can't execute 64-bit code in compatibility mode.
And the second, i understand it allows us to execute 64 bits code, what are avantages to use one or other ?
That question has been answered many times. Have you already tried asking Google
here is specified amd processor does x86-64 intel has the same working ?
Yes. (There are a few small differences between AMD64 and IA-32e, but the GDT is not one of them.)