Irish rights in the UK were admittedly woolly but our basic position was set out in the Ireland Act of 1949 which said we weren't foreign.The right to work does not predate the GFA by a lot. It was a very contentious issue for a long time. It still may be for citizens of the RoI.
That right is implicitly guaranteed in the (rather short) text of the GFA as both governments "recognise the birthright of all the people of Northern Ireland to identify themselves and be accepted as Irish or British, or both, as they may so choose, and accordingly confirm that their right to hold both British and Irish citizenship is accepted by both Governments and would not be affected by any future change in the status of Northern Ireland."
As NI natural can hold both nationalities they can avoid any legal barriers to employment and are not subject to restrictions. This however does not avoid problems such as delays on border crossings that do place practical difficulties to people working across the border. But prior agreements that are not threatened (at least by the UK) also keep that freedom of movement in place.
The UK leaving the EU by itself in no way undermines the provisions of the GFA.
As for alleged WTO rules, that is false. It it were true then countries such as Canada or Japan (or indeed the EU) could not do "trade deals" either, favoring certain countries over others. Controls can be "eased if a deal is done, and a deal for these issues can very easily be signed. It hasn't already because the EU wants to extract far more from the UK.
As for the UK keeping troops in a territory that is an integral part of the UK, I don't see how that is a problem. But then again I'm big on states being sovereign on their own territory. The RoI has already relinquished its stupid claim over the territory hasn't it?
'(1)It is hereby declared that, notwithstanding that the Republic of Ireland is not part of His Majesty's dominions, the Republic of Ireland is not a foreign country for the purposes of any law in force in any part of the United Kingdom or in any colony, protectorate or United Kingdom trust territory, whether by virtue of a rule of law or of an Act of Parliament or any other enactment or instrument whatsoever, whether passed or made before or after the passing of this Act, and references in any Act of Parliament, other enactment or instrument whatsoever, whether passed or made before or after the passing of this Act, to foreigners, aliens, foreign countries, and foreign or foreign-built ships or aircraft shall be construed accordingly'
Troops in the north isn't a problem but the GFA committed to normalisation so sending troops back to protect border posts would be a problem.
Yes we gave up our territorial claim on the north as part of this. 94% yes in the referendum.