1, Absolutely! Just make sure to use the right wires, as in use 5V (Red) and NOT 12V (Yellow).
2, Maybe! Depends on how much power you can safely draw from a usb/internal usb port, can change depending on motherboards/an when they where made.
But older usb 1/2 ports are often limited too 500mA, early usb3 ports is often 900mA, and then there are often specific "charging ports" on a bit never motherboards that started a while after usb3 was starting to get used and they are often 1500mA or 1,5A.
Higher end motherboards did often up that a bit to 2000mA/2A.
USB3,1=1500mA/1,5A, USB-C=3000mA/3A or 5000mA/5A BUT at 20V (so no good for the mister).
And the "charging port" does in many cases need drivers installed in windows (do not know if linux/etc ones exists) and "activated" to get more then 500mA, one board i had the "activation" turned of after 3 hours i think it was (never used it so never really looked at it).
The power brick that comes with the DE-10 nano is 5V/2A and do work.
BUT after adding SDRAM/usbhub/wifi/maybe dongels for a keyboard and a controller then the 2A is not enough anymore, sure it works for some but strange bugs/etc can creep up because the FPGA does run on too little power.
So 3A-4A is recommended!
And then USB is not that good of a option anymore since not many usb ports can manage over 2A.
It sounds like you will have a computer and the mister in the same case and sharing a PSU is that right?
If yes then there are more questions/tips here!
1, Depending on powersupply if going the sata/molex way and motherboard+PSU if going the usb way.
You could maybe power the mister without turning on the computer, first and foremost it depends on how high standby-power your PSU can deliver.
Not really sure here but i think over 2A is very unusual, secondly if going usb way how much the motherboard can deliver trough ports/charging ports when off, again probably unusual with higher then 1-2A.
2, You will most likely need a powerswitch on the 5V line going to the mister (or unplug/plugin all the time) since it will have 5V on it all the time even if the computer is off, standbypower 5V that is and is a big chance that it will be too low A to run the mister fully.
So you will probably need a cable going out to a switch then in to the computer case again, or make a spot/hole/etc where a switch can be mounted and operated from the outside.
3, I have read that running the mister on too low power can damage the FPGA in the long run, if that is true or not i do not know, but why change it?
4, Is other people going to use the computer/mister?
If yes then it will probably be a good idé to idiot proof it someway, how i do not know since i´m not stupid enough to come up with all situations a moron can get himself in
