How to read Operational Amplifier documentation
Read carefully, that 250 MHz BW is when Av = +1 meaning the opamp is used with feedback and the circuit has a gain of 1, it buffers the signal. It is the frequency at which that gain of 1 ( 0 dB) has dropped 3 dB so the gain at 250 MHz will be - 3 dB.

This is an absolute value, not a difference. In practice the average of many opamps will be more than 250 MHz so that the manufacturer can guarantee that the value is almost always at least 250 MHz.

But as a designer I would not use this opamp if I must have that 250 MHz in my application, I'd use a faster opamp to get some margin and possibly change the feedback around the opamp such that I get a predictable bandwidth of 250 MHz (or whatever I want).

That the curve starts to drop around 100 MHz is meaningless, what is important is how much drop it has at a certain frequency. Almost everyone uses the -3 dB point as the interesting point. -3 dB because when the signal is fed to a resistor, at -3 dB the power will be halved.

Also note here the value of Av = 2. Since at Av = 1 and Gain times BW remains constant, the expected BW will be 125 MHz. Note (first graph) that the amplitude of the signal matters, at higher amplitudes the BW will be lower. Also the supply voltage matters and also other parameters. That is what these plots show.

A negative gain simply means that the opamp is used in an inverting configuration. The signal is inverted or "upside down". For a single frequency signal you could also say it is 180 degrees shifted in phase.

I'll say it again: read Opamps for everyone !

It is the proper introduction to everything related to opamps.