For movement of people, this is easy.
It's not necessarily easy, lots of corner cases. I had some business visitors from India last week; they needed to get a Schengen visa to visit France and a UK visa to visit me. If the UK is allowing open access via Ireland, we are effectively saying that Ireland has the authority to let whoever it wants into the UK. Hardly the "taking back control of our borders" that was promised to Brexit voters.
If the UK expels someone (e.g. a potential terrorist), what is to stop that person simply re-entering the country via Ireland? Is there a legal process by which Ireland can deny them entry? Does the UK simply tell Ireland "Don't let this person into your country"? Are we going to pay Ireland to run border control on our behalf - can't see them doing it for free?
Consider the situation with the Jungle at Calais. If I am an illegal immigrant/ genuine refugee/asylum seeker and I make my way across Europe, hoping to gain entry to the UK, do I go to Calais and try to get across the channel, or do I get the ferry from France to Ireland and simply take a bus ride across the border, knowing that neither the French nor Irish governments is that bothered about stopping me and providing me with a home and so on. Can't see British tabloids putting up with that for long.
OK, there's a lot in your post. Your main point seems to me to be that this is all very complicated. I agree. And everything would have been easier had the UK voted to stay in the EU. I agree with that too, and was disappointed the vote went the other way.
To answer your point on your Indian visitors, in order to visit Ireland, they would need an Irish visa (as Ireland is outside Schengen, the Schengen visa would not apply, as now). Under my (!?) scheme, yes, they could walk across the border to Northern Ireland and be inside the UK. Agreed, this would not be the full control of borders that a swivel-eyed Brexiteer would like. But they would not be allowed to work (legally at any rate), so I'm sure it could be dressed up as a degree of control having been maintained. The only way to have full control would be to put up a border and that ain't gonna happen. The issue of control was really about free movement of EU citizens. They are all going to be able to come to the UK anyway.
Your terrorist example is interesting. It would apply to all deportees, actually, and applies now. If the UK deports a criminal after the sentence is served, there is currently nothing to stop the criminal applying for an Irish visa and coming back. This also applies in reverse for deportees from Ireland coming to the UK. I don't think there would be any change from the current arrangement whereby the two countries share information. The lucky thing is that Ireland is outside Schengen. If it joined, then things would be more complicated. (Good question on the Calais Jungle - I don't think the ferry companies will be willing to let them on the ferry, as they will still need a valid visa to get into Ireland. It's not happening now. And remember, even if the illegal immigrants made it to the UK, it is their desired destination, so they are not going to form big camps in Armagh, so it won’t be the big visible problem we have now.)
The trade issue is also a thorny one. I agree with you that barrier-free trade with the EU would be great. I'm actually more concerned than you are that this might not happen, because for me the holy grail would be trade conditions as we have now! Free trade agreements with other countries could indeed cause a big problem for trade with the EU. In my opinion the bottom line is that the UK will not be able to sacrifice EU trade for an agreement with another country. So they won't happen if they cause problems.
One thing that comforts me is that big business is strongly in favour of the status quo. They will bring a lot of pressure to bear on the politicians. The EU does have form of making political decisions that cause a lot of misery (the Euro springs to mind), but not if industry is against it. Business was strongly in favour of the Euro because it reduced costs. All the politicos (and most of the civil servants) will have one eye on the lucrative retirement jobs they are all looking forward to, so it’s in their best interests to toe the line (if you know what I mean…).