OK, this is all so familiar to a Brit who moved out of the UK but who still works for a UK Company.....and then there's the E106..
as far as that is concerned have you seen this form?
http://www.hmrc.gov.uk/forms/ca8454.pdf
Forget DWP. From bitter experience nothing throws their call centre staff quicker than telling them you "work in the UK but live abroad" - it simply doesn't compute, their computer says "no" and you
will get shunted from one operator to another.
HMRC can be
almost as bad unless you talk directly to the people at the "Centre for Non-Residents" in Newcastle...they deal on a daily basis with the likes of crew and don't find our tax/Insurance circumstances completely outlandish...You might want to start by ringing them - there's a number on the form and they are pretty good over the phone.