Ra's Al Ghul reads: Loyalty. Your locations are not unique.
Characters without the League of Assassins team affiliation have no team affiliation and cannot use payment powers.

Lazarus Pit reads: Lazarus Pit is not unique.
KO Lazarus Pit -> Recover target stunned League of Assassins character.
Activate -> Target stunned League of Assassins character cannot be KO'd this turn.

Dr. Light reads: Activate -> Stun target character with a cost less than the number of Fearsome Five characters you control.
Boost 3: When Dr. Light comes into play, put all Fearsome Five character cards from your KO'd pile into your front row.

In an Evil Medical School deck (old school), say if I swarm the field on turn 8 and it is my intiative, and I'm facing a League of Assassins deck. If my opponent recruits the 8drop Ra's Al Ghul, I believe I can stun it with Dr. Light's effect as soon as Ra's Al Ghul hits the field and before Ra's Al Ghul's effect comes into play. Correct me if I'm wrong.

The main question is...
If he KO's Lazarus Pit to recover Ra's Al Ghul, can I in response of my opponent KOing Lazarus Pit, somehow use Press The Attack (assuming I have 4 exhausted guys) on my Dr. Light, ready my Dr. Light from Press The Attack and stun Ra's Al Ghul again with Dr. Light's ability? Or does Ra's Al Ghul text come into effect and negate Dr. Light's effect?

Thanks.

"In an Evil Medical School deck (old school), say if I swarm the field on turn 8 and it is my intiative, and I'm facing a League of Assassins deck. If my opponent recruits the 8drop Ra's Al Ghul, I believe I can stun it with Dr. Light's effect as soon as Ra's Al Ghul hits the field and before Ra's Al Ghul's effect comes into play. Correct me if I'm wrong."

That is incorrect. You can stun any character in play before Ra's recruit effect resolves (which means he's not a legal target since he's not in play yet), but once Ra's recruit effect resolves, his effect of taking away team affiliations is not able to be chained to. There is no point of priority between when Ra's resolves and his effect takes place, no chance to stun.

His effect is a continous modifier so it does not use the chain That is why, as Ryan pointed out, there is no priority point for you to chain to the effect.

Spot's Knight