Answers to your questions:
1) Please describe the noise. Is it low frequency hum or high frequency hiss?
2) It is normal for it to heat when it produces output power. But its maximum output power is low so it will get hot only when it is overloaded with a supply voltage too high or a speaker impedance too low.
4) The sound is very smooth if the output level is not too high which produces clipping.
What happened to your LM386? It is still available but maybe not for long because Digikey says, "Last time buy".