Can I leave all my drives connected, plug in a seperate SSD through USB, boot into Nobara live and install on that drive without it affecting my mint install?
Yes. Just double-check every part of the install process so you don't write to the wrong device.
Also, if I do that will it put the EFI file on the seperate SSD?
Probably yes (depends on the options you pick during the install process). The external drive will get its own boot partition with appropriate EFI files. Then to boot from it, you would select the external drive in your UEFI.
I use rEFInd as my EFI bootloader: It lets me chain load other boot options (external drives) without touching my motherboard UEFI settings. I leave it installed to my main boot partition, but it scans for other bootable partitions at startup. Then it auto-populates a selector list of my main install, or whatever other external devices are plugged in. It can chain load GRUB, other EFI bootloaders, Windows, etc from these devices, so you don't have to worry about compatibility with whatever bootloader the OS expects to use.