The criticism was primarily about the morals of the case.
Each claimed the allowance in full. So two claims were being made where in any other circumstances there would only have been one.
That's not correct, each claimed slightly more than 50% of their allowance as set out in the report, which is also clear that there is nothing immoral or wrong about that.
So they basically each claimed the full allowance as an MP, despite the fact that they were living together as husband and wife and any actual expenditure was shared.
No, wrong for the same reason as above.
This point is made in the conclusions, together with the recommendation that the rules should be amended. The suggestion being that in the case of two married MPs one should be able to claim the full allowance, the second should be abated.
The conclusion was that where MPs share accommodation, they should not claim 100% of their allowance each, but it is also said that it is not as simple as saying two people should only share one allowance because life does not work that way. So the recommendation is that MPs in such a situation should not each claim 100% of their allowance - as that was exactly what Balls and Cooper had done it was basically an endorsement of their approach, in an attempt to make sure that other MP's in a similar position followed the same approach of moderating their claims.
Both must have known full well that they were claiming twice the amount owing to a loophole in the rules. So whilst it might not technically be in breach of the rules at the time it was certainly immoral, and very hard to find any real excuse once it was revealed.
If you actually read the report you can see full well the lengths they went to try to make sure they were behaving legitimately by consulting with all the relevant authorities along the way. It sounds really painful tbh.
Which is why, as I understand it, they repaid one set of expenses, something in excess of £40k.
So yes they did do it.
I don't recall that and I can't find anything to say that is the case. It also doesn't fit the facts as what they did in respect of their shared claims seems entirely respectable.
The complaint made against them was not about claiming double, it was about a technical issue as to whether the new London home had to be declared their primary residence because on average they spent more nights there although it wasn't what they thought of as their family home. If that was the case, they'd have been entitled to claim on the Yorkshire home which they had owned for longer as their second home (and would have avoided capitals gains tax on their previous London home, which they said would have overall benefited them financially).
The report into MP's claims states that the only issue was a overpayment of £1363 each in respect of mortgage interest which they did repay.
https://publications.parliament.uk/pa/cm200910/cmselect/cmmemest/348/348.pdf
It's worth running through the list to see how overblown the whole expenses scandal was. With a relatively few very dishonourable exceptions most of the issues identified are small - simple administrative mistake in a pretty complex system of rules is a credible explanation for those on the face of it. That's not to excuse some of the exceptions which were shocking, but the way it's been used to tarnish all politicians as a class isn't really backed up by the overall findings.