The left fan comes on first, and whenever the AC compressor is running, so it's more likely to go bad than the right fan because it's used more. I had the left fan fail first, and it was an easy replacement with the exact same part from Amazon for $150. Shortly later, my right fan stopped working, and that turned out to be corrosion in the electrical connector for the fan. The fan worked great after replacing the connector. So you may want to check the connectors.
In terms of engine damage, doesn't sound like you did any damage. It's usually the people that keep driving when the coolant lets go that damage the engine. But if you want to know for sure, you can do a search on YouTube for diagnosing head gasket failures and valve issues. The Lambo is the exact same for diagnosing head gasket and valve issues as any other car engine.