To the legendary pkmn master: I never looked or checked most of my Pokémon's nature, and I still don't know my purified Lugia's nature, so I can't help you much there.
But what I probably did was.. well, I tried first by saving right before getting on the elevator to Master Greevil and then restarting for.. hmm, I think because I either accidently killed one of the shadow Pokémon, or it was too hard. But the team I went with to stay was leaving a Pokémon in storage, though I forgot who so I had an empty slow in party for Shadow Lugia, whom I caught with my Master Ball. Then when I had to fight Greevil, I changed to Lugia with either Espeon or Ninetales to do the weakening. I believe it was Espeon, but who knows. So I probably used Lugia to tank (since he was using Shadow moves on Lugia more) to heal my regular Pokémon, but also attacking.
For me, the Legendary Birds weren't TOO hard to capture. Some difficulty, but not much since they attacked Lugia mostly.. and I had Espeon and Ninetales, along with Ampharos probably. But the ones I had problem with were Rhydon and .. the other one he had, Tauros. But I managed to weaken them to at least/most to yellow and caught them with Ultra Balls. Actually I think one was with Great Ball because they refused the Ultra Ball. Lol, that was weird, buuut, I wouldn't complain. ;p