Yes they do, but it's not a big part of your credit score.
Mine says, "no entry on the electoral register" and suggests my score will improve if I register.
Can't answer for tax allowance, as I'm UK resident, but it's 20% tax on taxable pensions not 25%. Suggest you use a different pension advisor if they can't even get the rate right.
Really better to stay with a UK address, no proof of life forms, no problems with credit checks, can even use your last UK address as nobody ever checks (even if mail is returned).