The clipping behavior is a bit surprising. I would expect it to flatten and then reverse and shoot to the positive rail as the input approaches negative rail, not to stick towards the negative rail before hard clipping.
It was unity gain?
As for the fakes, sure, all popular opamps on AliBay these days are fakes because the sellers ran out of harvested ones long ago.
More obscure ICs can sometimes be found genuine used
Decap them and see what's inside. Probably some LM358. Check bias currents.
They really don't bother faking those parts accurately. Almost all NE5532 are actually some unidentified Chinese RC4558 (see Zeptobars), nobody cares about wrong polarity of bias current and lower bandwidth. Similarly, I have bought some precision JFET opamps which turned out to be NE5532. The latter totally makes sense, those things are made for the audiophool market for upgrading NE5532s so nobody is going to notice that something is off.
The worst case output swing is specified at ±12V, with a bipolar 15V supply, with is 3V away from either rail, so the distortion is not surprising. The output stage of an op-amp is normal asymmetrical so the fact clipping occurs on the negative rail first is also unsurprising.
No, it is surprising. It has to be fault of the input stage, any funkiness in the output characteristics feedback would correct. Besides, if anything, it would be soft clipping rather than the opposite of it.
It's the input stage running out of CMIR, gain of 2x would fix it and give hard clipping at 1~1.5V above negative rail, by my estimation.
And it still is surprising. Normally what happens is that pulling the inputs to VEE shuts down the input stage and causes the output to jump up, pulled by VAS collector load CCS.