Is it possible to connect the GPIO's pins to a floppy drive? Not about the shape of the IDE cable, but about a voltage problem, may I ruin something?
Floppy and IDE interfaces are extremely different. I'll address the IDE as it is much more practical: GPIO-based communication with an IDE drive is most likely possible provided you operate at a low data rate and use passive level translation.
IDE is actually a very simple interface; it (and it's close cousin compact flash) have been interfaced to lots of legacy microprocessors, you just don't hear as much about that since the price on SD cards dropped causing most of those projects to be done with spi (or bit-bang spi) to SD cards instead.
Additionally, if not enough I/O lines are available to talk to the 16-bit IDE interface, it's actually possible to use only 8 data lines, since the commands are all 8-bit. Not having the other 8 bits means you waste half the storage capacity, but disk is cheap, so that's not necessarily a problem unless you need to read a disk created by something else. (You can however copy a normal filesystem image to or from such an 8-of-16 disk connected to a normal linux box, by manipulating the devices as a file)
Possible to connect some pins? With suitable wiring (maybe buffered), yes.
Will it do anything useful? Probably not. Maybe you could step the head or something, but transfering data would require a pretty complex controller. It would indeed be better to go with a USB fdd if you need to read/write floppies.
Pretty much all floppy drives are also (mostly) Shugart cabled, not PATA (IDE).