The position on the EU is false. The thinking is that they would have to petition the EU for entry like any other new nation.
It's all well and good to say Scotland will keep the pound, they are absolutely entitled to do so as it is a freely tradeable currency but I highly doubt the BoE would even consider being the lender of last resort to a country to which it has no policy influence. Without that Scottish paper will be worthless. They would need to start their own sovereign bank and peg a new currency to the pound and call them McPounds or something.
The border stuff is silly because they ignore that they would have to petition to join the EU and sign up to Schengen, the former is only possibly with the latter in hand. If Scotland signs up to Schengen while Britain doesn't, it does mean border checks. Britain doesn't have an open border with Schengen nations, only Ireland.
The point about North Sea Oil is just outrageous, plus it also counts Orkney and Shetland who's residents have previously said they would remain part of the UK if Scotland decides to become independent. Unless a new Scottish government is going to force O&S to remain in Scotland (I don't see that as feasible because the UK will not allow it) I don't see the oil and gas figure as being anywhere near true. In addition, the Scottish government has been much less favourable to tax credits and investment breaks for the oil and gas industry than the UK government, currently there are all kinds of decommissioning tax breaks which has allowed for money to be spent on developing new fields and getting more out of dying ones. That cost the government £3bn last year and £1.5bn this year, but next year the new fields and new output will benefit the Treasury by £2bn and then £3bn after that for 3-4 years. Scotland would find it tough to finance that £4.5bn upfront cost without making severe cuts to the state finances or borrowing an untenable amount of money. The money doesn't sound like a lot in isolation, but in relative terms it would be similar to the UK borrowing £45bn to give away in tax breaks for a couple of years with no guarantee of returns. £4.5bn is a small bet for the UK but it would require a Scottish government to leverage the nation's finances and hope it works out.
I'm not saying that Scotland can't be independent, far from it. I just think it is absolutely ridiculous for the SNP to state that everything would stay the same when Scotland would have a massive, massive deficit (>10%, including all oil and gas, 15% including only Scotland's share) and no lender of last resort to back any paper. That means huge budget cuts all over the place. Free university would be the first to go, second would be free prescriptions, then there would be new alcohol taxes, corporation tax would go up (then come back down as companies run for the border), income tax would go up. The scenario painted by the SNP - that independence means no change - is a dishonest one, and sadly anyone who points that out will be labelled a hater or "talking down Scotland".