You can go to NI without a passport. However, if you are flying you need to make sure that the carrier accepts a photo drivers licence as valid ID. I think that the rules for the main carriers (Ryanair/Air Lingus) differ, so have a look on their websites. This is not an immigration issue, but an airline issue.
The latest that the application must arrive at the Home Office is the day that the visa expires (which differs from other leave to remain applications). You must remember that if the date of expiry is a weekend or a holiday that the application arrives on the previous working day.
There are no restrictions in terms of re-entering the UK. Assuming you do not need a schengen visa there should be no restriction to travel, but this really depends on the immigration laws of those specific countries.
No issue with the passport.
Vicky