How important is to connect the diode's metal case to the ground for
its protection? Should I buy a negative -5 V voltage regulator?
Employing a negative voltage regulator is the safest route when the laser diode anode is galvanically connected to its metal body. If you fed the anode from a positive bias generator and the metal body got grounded by a user making an optical connection to the laser, it would short out the laser (not harm it of course) and short out the bias. However, if this isn't seen as a big deal then use a positive bias to the anode.
If this might be a problem then use a negative supply and bias circuit as per what you posted in a comment: -
You should reference the place that you found the above circuit to abide by SE rules BTW. Anyway, the circuit looks good to me except I might employ a higher value base resistor. You might have to mess around with the value of C1 and R2 in case there is instability. The op-amp you picked is quite low bandwidth (200 kHz) so the integration circuit might need C1 increasing maybe 10 to 100 times. The op-amp is also only capable of a 20 mA drive current at the output so it might just run out of steam as you push the laser current towards 500 mA.
Or, you might decide to use a low-gate-threshold-voltage MOSFET instead of the BJT.