Mostly power out. An op amp is designed to go to frequencies above audio (even if it doesn't go extremely high frequencies), while a LM386 is a specialty part where it doesn't matter.
An op amp is much simpler to predict mathematically. A LM386 is meant to be a power audio amp, which it does relatively well. Nowadays there are better, but not cheaper or as common.
No.
Many older opamps cannot produce upper audio frequencies.
The LM324 and LM358 opamps have a max output that drops at about 2khz.
The 741 opamp has a max output that drops above 9kHz.
The TL07x opamps have a max output that drops above 100kHz.
The inputs of the LM386 are biased at 0VDC, not at half the supply voltage like opamps are. When the inputs are at 0VDC then the output is at half the supply voltage.