± usage and absolute sign ambiguity
You are almost right.
One other very important characteristic is the common-mode range: In conventional opamps this must not get closer to the supply rails that ~2V.  For the NE5532 it is shown as below. Note that when supplying +/-15V the device is only guaranteed to work if the inputs are in the range of +/-12V. A typical device will go about a volt further than that to +/-13V.
The device won't be damaged if either input goes to either the positive or negative rail but it won't function as an amplifier. In fact with some devices, the gain may actually reverse and the negative feedback will turn positive causing latchup.
Most opamps of this type have traditionally been used with dual supply rails so the convention is is to describe the supplies as +/-. You could use it as a single supply but the inputs and outputs wouldn't be able to go to ground.
Some amplifiers, commonly referred to as single-supply devices do allow the inputs to go down to ground. Others referred to as rail-to-tail allow signals (input, output or both) to go to either supply and the device still operate. There are often compromises in the operation of such devices though so you need to study the datasheet catefully.
Also be very careful about the distinction between the maximum ratings where the device may suffer damage and the values shown in the characteristics table where the conditions required for normal operation are described