Why would you start the axes at zero? First off, as you note, the income axis is logarithmic, and so cannot go to zero anyway. As for life expectancy, zero would be a meaningless label. It's impossible for a country to have a life expectancy of zero. It is entirely appropriate to set the minimum value for an axis at the minimum value which has ever been recorded. The difference between a life expectancy of 40 and 75 is enormous, and I do not find the presentation to be in any way misleading.
Your second issue, the logarithmic axis for money, is debatable either way. Given that incomes have generally risen exponentially (in the US, an increase of about 2% per year for the last 200 years), a linear scale would show accelerating income growth for wealthier countries. It strikes me that this would be more misleading than use of a logarithmic axis. If you usually think of income growth as linear, maybe it's your thinking, rather than his graph, which is mistaken.
For the third issue, there is something called "Purchase Power Parity" which corrects for the effect you're talking about. The presentation doesn't discuss whether his income figures are adjusted for PPP or not. Contrary to your assumption, the figures clearly are at least adjusted for inflation (given that his $400 minimum would have been a princely sum in 1810, far above any country's per capita average), and if he's adjusted for inflation, I see no reason not to believe that he's adjusted for PPP as well. If he hasn't adjusted for PPP, then I agree that's something that should have been done, but it in no way alters his fundamental point. PPP reduces income inequality, but in no way eliminates it.
For the fourth issue, without his enthusiastic presentation, it's just a graph. There's a time and a place for cold, sober, "just the facts" presentations, and that is textbooks. In less academic settings, it's entirely appropriate to use enthusiastic explanations to show people why something matters.