> What does efficiency in tax mean?
In this context of optimal tax theory, it'd be referring to the "deadweight loss" of the tax (how much economic activity is reduced per dollar of revenue), and to how much it distorts behaviour. (When viewed another way, it's a measure of how easy the tax is to dodge.) Generally speaking, land taxes are the most efficient, followed by consumption taxes, income taxes, and finally taxes on capital are the most distorting.
> VAT is regressive imho
Correct. However as a general rule income redistribution is achieved much more effectively via spending than taxation. The most efficient, effective welfare states have relatively regressive taxes skewed towards consumption taxes (which as above are efficient and hard to dodge), and then use the revenue to fund programs targeted at the poor. By contrast the US uses very progressive taxes to fund programs largely targeted at the middle class. Empirically, this model doesn't work very well. :)
> Capital gains tax ought to be the heaviest
Capital gains is the least effective tax. It distorts behaviour, has high deadweight taxes, is the easiest tax to dodge, the economic burden is seldom felt by the intended targets. It's not good at redistributing income, and it's not good at raising revenue.
I live in a social democratic country with a strong welfare state funded by a fairly high VAT, with no capital gains taxes at all. It works quite well.
The US, in my view, has managed to get itself into a state where it tries to use the tax code to solve all social and economic problems. Instead of universal health care we have tax deductions for employee provided insurance, and subidies and penalties for individual insurance, handed out via the tax code. instead of welfare payments we have EITC. And when it comes time to look at at inequality, the discussion starts (and stops) with an argument about how progressive the US tax system is. At this point almost any reform will make the US tax code more regressive; the trick is to realise that's okay. Social democratic states work best when you use the tax code to raise revenue, not bring about utopia. But that's a rant for another time. :)