By "faulty", I mean that you can have errors on the ancilla qubits, you can have faulty syndrome extraction, etc.
This is the paper I found really convincing once I’d worked through it: https://arxiv.org/abs/quant-ph/0504218 This was really the start of rigorous threshold proofs, and is the starting point for a number of subsequent improvements.
This paper by one of my group mates includes faulty prep on ancilla qubits and other errors in an analysis of Bacon Shor. I'm not sure what you mean by faulty error extraction (measurement errors?) but it has an explanation of the error model.