Do immigration checks exist between Northern Ireland and England/Scotland? (not just for flights but also ferry/bus/train)
Do immigration checks exist between the Republic of Ireland and England/Scotland?
All of the areas you mentioned are in the so-called Common Travel Area...
The Common Travel Area (CTA) is a travel zone that comprises Ireland, the United Kingdom, the Isle of Man, Jersey and Guernsey. In general, the CTA's internal borders are subject to minimal or non-existent border controls and can normally be crossed by British and Irish citizens with minimal identity documents.
The UK Border Force does not carry out routine immigration checks on travellers (regardless of nationality) arriving in the UK from another part of the CTA. However, because the Channel Islands are separate from the UK for customs purposes, it carries out selective customs checks on travellers arriving from there.
So there are no checks other than random checks.
The checks for people arriving from within the Common Travel Area to the Republic of Ireland by air or sea have some formal arrangements, but they are also largely random...
In 1997, Ireland changed its immigration legislation to allow immigration officers to examine (i.e. request identity documents from) travellers arriving in the state from elsewhere in the CTA and to refuse them permission to land if they are not entitled to enter. Although this is stated to apply only to people other than Irish and British citizens, both of the latter groups are effectively covered as they may be required to produce identity documents to prove that they are entitled to the CTA arrangements.
NOTE: although arrivals from the Irish Republic to the UK are not subject to inspection, there are some conditions where the person becomes an illegal entrant.