Don't think it's correct the EU wouldn't want Scotland, I think they would do all they could to have an independent Scotland rejoin.
The border is another issue but let's face it the UK can't be trusted in anything it says at the moment so we wait for a somewhat trustworthy government to come to your country.
The EU isn't an all benevolent entity, it makes fuck ups like any administration but on the whole in my experience it is more trustworthy than most national parliments but it is still a political machine at the end of the day and they tend to want to grow as big as they can.