It is odd, isn't it? The effect sizes seem ridiculous*, but there's nothing obviously wrong with that study (aside from the sample size). Cochran has blogged about oxygen before as well. To compile some of the relevant papers:

• Winter et al 1975, "The anesthetic effect of air at atmospheric pressure"
• Scholey et al 1999, "Cognitive performance, hyperoxia, and heart rate following oxygen administration in healthy young adults"
• Kajtar L, Herczeg L, Lang E. 2003. "Examination of influence of CO2 concentration by scientific methods in the laboratory" In: Proceedings of Healthy Buildings 2003, 7–11 December 2003. Singapore:Stallion Press, 176–181.
• Kajtar L, Herczeg L, Lang E, Hrustinszky T, Banhidi L. 2006. "Influence of carbon-dioxide pollutant on human well-being and work intensity" In: Proceedings of Healthy Buildings 2006, 4–8 June 2006, Lisbon, Portugal:Universidade do Porto, 85–90. (requested these two)
• Chung et al 2008, "A study on the effects of 40% oxygen on addition task performance in three levels of difficulty and physiological signals"
• Satish et al 2012, "Is CO2 an Indoor Pollutant? Direct Effects of Low-to-Moderate CO2 Concentrations on Human Decision-Making Performance"
• Law et al 2014, "Relationship Between Carbon Dioxide Levels and Reported Headaches on the International Space Station" (I wonder if the ISS has the same kinds of 'sick building syndrome' that terrestrial buildings do and so the relationship may be confounded? On the other hand, the 'chits' mentioned sound like quasi-experiments.)
• Stafford 2015, "Indoor air quality and academic performance" (Texas natural experiment)

The problem for me is that while it makes sense that since we run on oxygen and the brain uses a lot of oxygen (the whole 'BOLD' thing etc), more oxygen might be better, it has the same issue as Kurzban's blood-glucose/willpower criticism: if the brain needs more oxygen than it's getting, why doesn't one simply breath a little more? While sedentary during these sorts of tasks, you have far more breathing capacity than you should need - you are able to sprint all-out without falling over of asphyxiation, after all. So there's no obvious reason there should be any lack, even more so than for glucose. And shouldn't CO2 levels closely track various aspects of weather? But as far as I know, various attempts to correlate weather and cognitive performance or mood have turned up only tiny effects. In addition, too much oxygen can be bad. So is it too little oxygen or too much nitrogen or too much carbon dioxide...?

Jessica Taylor for lending me a CO2 monitor so that I could see variability in indoor CO2 levels.

What monitor is that? You could try recording CO2 long-term, especially if it's a data logger. Opening windows is something that's easily randomized.

I did some looking and compiling of consumer-oriented devices a while ago: https://forum.quantifiedself.com/t/indoor-air-quality-monitoring-health/799/40 I was not too impressed since nothing hit the sweet spot of accurate CO2 and PPM measurement under $100. The Netatmo looked decent but there are a lot of complaints about accuracy & reliability (checking the most recent Amazon reviews, still a lot of complaints).

I've been thinking maybe I should settle for the Netatmo. I've been working on a structural equation model (SEM) integrating ~100 personal data variables to try to model my productivity (some current sample output), and it would be nice to have even noisy daily C02 variables (as long as I know how noisy and can use it as a latent variable to deal with the measurement error). Correlation-wise, I think backwards causation can be mostly ruled out, and the most obvious confound is weather, which is already in my SEM.

* taken at face value, with reasonable estimates of how much rooms differ from day to day or week to week, CO2 levels would explain a lot or maybe most of variability in IQ tests or cognitive performance!