But it's not yet clear whether Scotland becoming independent would count as a new state joining the EU, or whether it would count as a successor state and inherit all the UK's various opt-outs (to do what it wishes with).
Wrong! Both the EU (and NATO for that matter) have said that residual UK would be the successor state and that an independent Scotland would have negotiate membership with all other member states approving (and that, for political reasons, is not a given eg Spain might well say no to try and dissuade Catalonia from going independent).
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Participation in Schengen is compulsory for new states joining the EU. It didn't used to be, but I believe this changed with the 1999 Amsterdam Treaty. All new states joining the EU after this date are legally bound to implement Schengen.
A similar scenario occurs with the euro, where new member states are legally bound to join the eurozone, if they meet the criteria to do so.
Absolutely correct.