UEFI System. Are EFI settings kept on the HDD. What happens if I change HDD?

EFI settings are stored in NVRAM, typically some flash on the motherboard (part of the flash chip which stored the motherboard's firmware, including the UEFI itself). See Wikipedia for more details.

So you can safely change hard drives without losing EFI settings.

During the Windows/Linux installation with EFI, the installer puts a startup file onto a special partition on the disk. This EFI partition is usually mounted under /boot/efi in Linux. The EFI partition (see also http://en.wikipedia.org/wiki/EFI_System_partition) is a FAT partition containing the OS's boot subsystem, e.g. /EFI/BOOT/BOOTX64.efi.

The EFI settings in NVRAM are read by the UEFI upon boot and contains pointers to the files in this partition. This acts like a boot manager with different options, like GRUB (albeit GRUB itself is sometimes called by it). Therefore an OS will add an entry in the EFI NVRAM for itself which will point to its files on the EFI partition, e.g. EFI/ubuntu/grubx64.efi. The efi boot manager can be configured under Linux with the efibootmgr command, or in your BIOS setup. In both you can also set default boot entries. Manipulating these entries may be more difficult with the "Secure Boot" BIOS option enabled.

In your case there will be invalid entries in the EFI boot manager (NVRAM area) once the HDD is removed. However, your new OS will install (and likely set as default) its newly installed option in the EFI boot manager.