No. EROEI does not attempt to account for externalities like climate change. It isn't part of the formula at all. Energy in, Energy out. That's what matters to it.
Lifecycle costs analysis may or may not include costs from climate change in the model depending upon who is doing the analysis. Some analysis is done to model the economic viability of a particular project, so it only cares about those costs that may actually impact the project (i.e., captured costs). That sort of analysis will by design not account for externalities because, by definition, those will not impact the project.
What the fuck are you talking about? He responded to a very specific insult ("Gretas") with a very specific answer that was totally appropriate in a fucking thread titled "Climate Change" not "Rex ignores basic science and how it impacts everyone."
Who do you think he was implying are the psuedo-scientists here? Because I can promise you Rex wasn't intending to dismiss the few phony scientists that dismiss human driven climate change. The vast majority of climate scientists understand how releasing carbon into the atmosphere changes the planet's climate.
EROI is a moving target. EROI of solar is higher as non-dispatchable renewables become a larger percentage of the grid due to the need for additional storage. But, up to 50% or so renewable, modern utility solar would have an EROI competitive with natural gas. There are a lot of different calculations out there for EROI with a lot of different assumptions.