No. dB is not a unit. If it's quoted as a unit, then the person using it doesn't know what they are doing. dBx where x is some reference value is a unit and that means dB relative to the reference unit. So dBm/dBv etc are units. dB is not.
Mathematically, dB is the ratio between two unit values and for example if you divide metres by metres you cancel the units.
Exactly. If someone says that "this plane is about 130dB" what they mean is in reference to 0dB given 0dB is the threshold of human hearing. It seems most people are confused by dB because they don't understand it provides a logarithmically scaled unit compared to a reference which happens to be quite useful for something like audio where our perceptual models are stacked with these types of curves.
No they will specify something like dB SPL, A-weighted with Fast time integration. Sometimes referred to as dB(A) SPL.
And measured with a particular method, which specifies things like distance from sources, environmental conditions, etc.
Though that's still incomplete, because (more of the context stuff that you just have to know already) the "A" here refers to the frequency-weighting scheme used in the measurement, and not to the reference level (which is SPL).
It should probably be given as: dB(A) SPL, or dB SPL (A-weighted).
The fact that dBs aren't units, but are used like units, is the point being made.