Is a strong foundation in advanced microeconomic theory essential to study environmental economics at the graduate level? Advanced theory of individual economic behaviour in production, consumption, and general equilibrium. Are these topics useful to study environmental economics a the graduate level, or is there not a lot of overlap?
It depends on what you like to do in environmental economics. There are lots of people who are working on moral hazard issues on environmental economics by principal-agent models. 

More specifically, an interesting question arises from non-point source pollution. Let's say farmers are using pesticides which cause pollution. Probably, you don't know specifically which of farmers are responsible. How could you create "incentives" in order to avoid this situation(pollution) ? 
You can look at the Segerson (1988), published in Journal of Environmental Economics and Management in order to understand this issue more precisely. 

Another field could be about social choice theory (and decision theory) in environmental economics, especially on equity between individuals and generations. These issues can be included in microeconomics but more in applied math as they use topology etc. (Geir Asheim, Stéphane Zuber's papers could be interesing for this subject.)

Just to finish, I think there are more things to do in environmental economics at a macroeconomic level. (Note that my opinion could be biased as I am working at macro level.)