You need a passport or an ID.
Yes, you will need to show your passport.
Yes, You always need a passport (or EU national identity card) to get in our out of the UK.
Is a swaziland passport need a visa to uk?
Yes.
Northern Ireland is part of the UK so you need no ID to travel there. In fact since the UK and Eire are in a common travel area so you need no idea to travel anywhere within the British Isles. However since the war on terror, many airlines will require you produce ID, However there were no such caveats during the troubles.
Strictly speaking, as the UK and France are both in the EU, you don't need a passport to travel to France. However, you will need a passport on your return to the UK. To be absolutely certain, you should check with the French Embassy.
As the "Common Travel Area" is in place, you do not need a passport to go to Dublin from the UK. When flying between the UK and Ireland it is best to ask your airline what form of ID will be accepted.
Absolutely !
a letter weighing 1 gram to Eire, what postage do I need?
No.
No, Not if you a Citizen of the UK or Ireland.