I’m too knackered to write a detailed tailored answer but I’m in a similar situation to you (slight difference may be that I’m a freelancer for the UK company, rather than a remote employee - not sure about you). Here’s a recent relevant thread of mine:
Briefly, you don’t need to set up a Taiwanese bank account but you should pay taxes to Taiwan not the UK, assuming you’re tax resident here (no idea how it would work getting out of PAYE in the UK if you are an overseas employee). You might want to open a local bank account anyway, but it’s not essential (or really relevant to your tax liability).
You would still be eligible for NHI, assuming the new rules (no six-month waiting period) have come into effect - see here, for example:
Yes, it could, but that would depend on the specifics.