What will happen if I by mistakenly connect GND and +5v of an Arduino Nano? Is there any protection for it? Or will it get short and fry out something?
Depending on how you are powering your board different things will happen.
If you are powering through USB from a PC the PC should, according to the USB standard, detect an over-current situation and disconnect the power from the port. However some don't, in which case see point 2.
If you are powering through USB from a USB power supply (phone charger, etc) the charger's output voltage will drop to near zero and the current will hit maximum for the supply. This may or may not damage the supply, but either way it's not a nice thing to do.
If you are powering from a small external battery (say a 9V PP3 battery) things will start to get hot. The voltage regulator will most likely overheat, the battery may overheat, but not much damage should occur - a small PP3 battery can't provide much current.
If you are powering from a large external battery (e.g., Li-Ion cells) or an external power supply, things will get more nasty. The on-board regulator will overheat. If it isn't thermally protected (genuine Arduinos should be, but cheap Chinese clones may not be) then it will fail and you will find smoke coming out of the regulator. At this point the regulator is dead. If the board contains a polarity protection diode that may fail in the same way (I have seen these blow clean off the board before). Physical damage (blackening and blistering) may occur to the PCB as well.
So all in all shorting +5V to GND is never a good idea.
That means it will heat up and shut down if over-loaded. Of course, your nano could use a less-effective regulator and simply smoke away. It's also possible that the 5205 would not clamp in time to prevent damage on an instant dead-short. Generally, i've found the arduino boards to be fairly resilient, compared to other MCUs, so you can probably get away with it at least a time or two.
Dead shorting is never a good idea even if you can get away with it because there can be incremental (as opposed to catastrophic) damage. You might cause instability in the regulator, and you might limit it's ability to limit volts. Those problems might not be revealed instantly, but instead at the worst (unexpected) time.
EDIT: based on your edit, it sounds like your "FTDI" is fried, but not necessarily your MCU.