With respect to the excessive negative input voltage, I think it might help if the input in question is the inverting input, and the feedback resistor from the output connects there too. As the input goes negative, the output goes positive, and the feedback resistor allows the "-" input to stay near the negative rail. Ideally, it's supposed to stay at 0V up to a certain point, but it doesn't really stay there.
I had the same issue driving an LM358 from the audio output of a walkie-talkie. Volume is supposed to be set to 1, but several times I've found it at 4 or 5, and so far the LM358 has survived. The input resistors are mainly responsible, but I think the feedback also helps. Circuit attached.
I should say that I'm not an EE, so if this is completely wrong, I hope someone will correct me.