Not any more—they are currently running at a net loss. the rest of the UK are subsidising them to be decommissioned and for staff to be made redundant.
True but that's been known for a while - and afaik there hasn't been a definitive "oil will run out in 20xx" answer. Main thing hurting right now is the global slump in oil price hammering the North East - to be fair all of this does suggest that Scotland needs to focus on diversifying into renewables and tech but that was always the plan. It's just kinda irritating that this has come up now as it's a convenient stick to beat the pro-EU left with.
All current economic statistics about Scotland are baed on origin. Anything made in Scotland is counted as part of Scottish economy, even if it travels to England to be exported—that is counted as a Scottish export even though it left via an English port.
The difference after independence would be that the export would occur at Carlisle rather than Felixstowe.
Most of the "Thatcherism effect" of the 19080s can be directly attributed to North Sea oil. If you remove that money from the economy Thatcherism was a contraction until the banks boomed (though that was mostly post-Thatcher).