Twiggy the Turtwig
Due to being a famous singer she must be female. Twiggy also doesn't like getting wet, so she must run from any Water type pokemon. If Twiggy is hit with any form of water she gets extremely angry and must use her most powerful attack (factoring STAB and resistances/weaknesses) until the perpetrator DIES (or faints...whatever). May evolve after soloing a gym leader of your choice, and again after soloing all of the lackeys of two other gyms of your choice.
Boo the Gastly
It doesnt like to be seen, so it can only attack while a pokemon is asleep. If the pokemon is awake it can only use Night Shade.
It must have this these 3 moves:
Hypnosis
Dream Eater/Nightmare
Night Shade
The fourth move is your choice.
It can evolve into Haunter after it solos Fantina's Gengar, or reaches level 40 if incomplete.
It can evolve into Gengar after it solos Candice's Abomasnow and Volkner's Luxray, or reaches level 60 if incomplete.
Augustus the Carnivine
Augustus hates the fact that it floats above the ground, so it must use Ingrain the first turn it comes in every time. His only STAB is Solar Beam (from a TM ASAP), at least until it learns Power Whip. After that, he must replace Solar Beam. After he learns Power Whip, he must also solo every Grass-type you encounter to prove it's superiority.
Myopia the Girafarig
This Girafarig is has terrible eyesight, so it must use Odor Sleuth in order to attack a foe. (Girafarig learns Odor Sleuth at level 5)
After it learns Baton Pass at level 23, it loves BPing so much it can never switch out, only Baton Pass.
Ford the Slugma
Named after the only president who was never elected; you must never place him in the lead position. Note that you cán lead with him whenever a fainted pokemon occupies the first slot.
In an attempt to solve the Watergate scandal, the reason of his sudden appointment, he must solo 10 trainer owned water types to evolve into Magcargo.
Because the Cold War was still going on he must learn a stat up move (Amnesia, Harden, Lightscreen, Substitute, Double Team.. etc). This is to keep up with the arms race that was going on at the time. But during his presidency there was also a certain relaxation in the political tension of the Cold War. To add to this 'détente' you must thaw any Ice Type you encounter with a super effective fire attack, to prevent the Cold War from escalating.
When he has evolved into Magcargo he still has an economic crisis to deal with. Solo Byron in an attempt to solve the prices of natural resources. You may use other team members to set up, but they must not attack. Should you fail, he must give up on one of his stab moves. (If he hasn't evolved at this point due to the water challenge you should solo any two trainers left to evolve)
Trollamus the Hippopotas
It likes summoning weather that no one else on its team likes, so it must be sent out first against every gym leader and E4 member not being soloed.