The concerns about LF's execution are misplaced. He DID have a trial, as much as anyone ever has on this show. He was presented with several charges, and admitted to at least two of them. Admitting to murder is enough to justify summary execution with no further process.
My wife and several friends are all unsullied, and when they're confused, they just ask me. They almost never look stuff up online because they fear spoilers. Not all ASOIAF spoilers are properly labeled and hidden.