The forum ate my post but I kept a backup copy. Why is this even necessary?
1 - Does that mean that it will be pulling only -8V from the available -22V?
Huh? No, with the 78L08 positive voltage regulator, the operational amplifier will be operating from a regulated +8 volts and an unregulated -22 volts which is no problem. Of course you need a 36 volt minimum regulator for this to work.
The +8 volt supply is really a little high and could be lowered. The Tektronix PS503A power supply uses similarly asymmetrical voltages for its operational amplifiers but only 5 volts and 78L05s are much easier to find than 78L08s. I have seen 3 volts used in some circuits.
2 - Wouldn't it be better to just install another 7908 regulator and feed -8V to the OP Amp?
The operational amplifier has to drive the adjust pin of the LM337 down to the -ve output voltage (plus 1.25 volts) which is considerably more negative than -8 volts.
The recommended OP Amp is LF351.
There are lots of parts which would work including the LF351, LF355, LF356, TL031, TL051, TL061, TL071, TL081, 741, 301A, etc. Practically any 36 or higher volt part is suitable. I have used the LT1007, a slightly more modern OP-27, in a similar circuit for low noise but that does not apply here.
Usually a slow operational amplifier should be used but the output resistance of the operational amplifier combined with C6 implements dominant pole compensation preventing oscillation. In that respect, a 301A with suitable external compensation would be a much better choice but any part will work.