The answer is for the U.K. to simply leave with no agreement and conduct widespread background checks on anybody police even think may be in the country illegally - and as a matter of course for anybody traveling from Northern Ireland to the rest of the U.K.
Those in Northern Ireland would have to accept this as the price of not having a hard border. That then leaves it to the EU to take the very unpopular step of building a border wall which the Irish will hate. (The UK’s concern is to control immigration while the EU’s concern is over the flow of cheaper goods not subject to all the EU’s regulations flowing from the U.K. into Ireland)
Yep. Agree, see #9 above.