Please be advised that updating the BIOS or UEFI firmware is only really required if the currently installed version contains serious bugs that affect the performance and/or stability of your system.
Just as an FYI, the UEFI version on my motherboard is about as old as yours. 
The reason why EFI variables are not supported is that your machine boots in legacy BIOS emulation mode, as the inxi snippet tells you.
You do have an EFI partition — which is strange, because if you install the system in legacy BIOS emulation, then it normally won’t create an EFI partition — but I suspect it will be unusable anyway, or at least, if you have a GUID partiton table (GPT), because in that case, the core image of GRUB will have overwritten the partition boundary of the EFI partition, and will have thus damaged anything that was in there.
This is why, when a GPT is used in combination with a legacy BIOS emulation boot, you must normally create an unformatted partition of about 2 MiB, of the type bios_grub. But if you have an MBR/MSDOS-style partition table, then this is not necessary, because MBR/MSDOS leaves enough room in the master boot record for the core.img of GRUB.
No, not normally. But best is to write down the settings of the currently installed one, so you can reapply them after the update, which will come with default settings.
I would recommend not updating the firmware unless you’re having problems with the current one.