Beta 13 comes with a 4.3 GPE side already - there is a small OTA update to that GPE rom that is yet to be included, this is the OTA notification you have been getting - it will fail to install even if you tried to do it.
Paul will be providing updates in the future, I do not know when - for a while his life involved around SWITCH and he had no time (aside from his vacation) to do anything other than SWITCH, I assume he is just catching up with other work - this, after-al is his full time job and he has commitments he has to meet so SWITCH has had to take a back seat for the time being.
Yes you enable NFC in the settings and then it should work - the sensor is just under the camera on the HTC One.
If it is still not working for you then you will have to try a full wipe and re-flash as there is no reason why NFC will not work with SWITCH (like I said before, I personally tested it on all versions and use it every day).
You will never have an NFC icon on your notification bar whilst using SWITCH (or most generic stock, or custom roms).
The N in the notification bar is a carrier change stock rooms don't have it, it doesn't mean nfc isn't working. I use nfc every day and I have always made sure it works before letting Paul release an update.
If it's not working it is something to do with your phone and/or setup and not the rom.
Try to untick "compression" in TWRP if you haven't already that fixed the error for me when I had it.
A week or so later I turned it back on and it backed up fine.
EDIT: Actually I am starting to think that maybe the files are too large for TWRP to be able to tar (downside of Switch I guess) - I might be wrong, has anyone been able to take a compressed backup in TWRP since beta 10?