As for border controls, the Schengen zone includes non-EU members Switzerland, Norway, Iceland and Lichtenstein. There are no permanent border controls between those countries and their EU neighbours, although there may be random customs checks.
It would be a superb irony if the EU insisted that rUK sign up to Shengen if they want to remain in the trading zone. They could easily do that but rUK would be unwilling to sign up to it under a Tory government.
Failing that two countries can reach their own agreement not to have border controls. That’s the current arrangement between the UK and Ireland. There’s no reason why such an agreement couldn’t be reached between Scotland and rUK.
Failing that the rUK can install their own controls at the border if they like.