My ds voted remain and his Dad and I voted leave. We respect each others point of view and our democratic right to vote as we saw fit.
Ds makes me tea, I wash his clothes....we have lively discussions about our choices, but that's it.
I don't think the entire government is in chaos; after all Belgium went for over a year without even having government, so the PM resigning from October doesn't plunge the govt into chaos, as there is not being for the moment a cabinet reshuffle.
Juncker can say what he likes, the Council of Ministers will decide, and already Merkel and Hollande are making friendly noises; Italy said before the referendum that they will trade with us irrespective of the EU, Poland still counts us as an ally, and they all have to work with us in NATO anyway. Juncker is sulking because he now won't get our seat on the UN security council, and I can't see France giving theirs up. I can't see that 27 nations are going to decide to stick tariffs up if they don't want to do so, especially as so many of their populace live in the UK.
Don't know enough to comment about N Ireland, but Sturgeon was determined to have another referendum (bit like the EU, go back and vote again until it agrees with what we want) anyway. It is entirely possible that the Scottish electorate will still vote to stay in the UK, given they would have to have the euro, and go to the back of the queue behind those countries already waiting to accede. They'd be twisting in the wind for 5 years.
I'm afraid I have to agree with your Dad about getting a grip.