The big problem is there is a disconnect between the pictures and the words. The words say "ARM Arduino" that means a 3V3 one like the Due or the Zero. BUT the pictures show there is a restive divider on the logic lines meaning it is meant to be run with 5V signals.
The module has a 5V pin, would be actually safe, for the SD card, input that voltage in there? Or shall I use the 3.3V output of my board?
It looks like that module has an on board voltage regulator so connect the +5V and do not connect the 3V3.
Is the MCU actually fully compatible with a 3.3V logic level for SPI communication? I mean, what if the MOSI pin receives 5V?
It looks like the module has potential dividers on the pins.
The pins of the module, are those labeled in pairs or just for a single one?
Not at all sure what you mean. Why should anything be in pairs?
Which function will put an exact copy of this array into the file? write or print?
Nether. But write puts the actual byte in the file and print puts an ASCII representation of the value of the byte.
The size count of the file is updated every time I write (not overwrite) a new byte into it, right?
Don't know.
The print function writes an unsigned int variable as a character string
Yes.
is there a way to convert an unsigned int type variable into an array of bytes? (in its "byte form")
Yes
upperByte = variable >> 8;
lowerByte = variable && 0xff;