For each category I have a few ideas on what I might play:
Board & Card Games
Board Game Top Shop (PS1)
Dokapon Kingdom (Wii)
Yu-Gi-Oh (one of them ha)
Baten Kaitos: Eternal Wings and the Lost Ocean (GC)
Sugoro Quest (Famicom)
Wacky PS1/PS2
Chulip (PS2)
Ape Escape games (PS1/PS2)
Devil Dice (PS1)
Moon RPG Remix Adventure (PS1)
ChoroQ (PS2)
Under the Skin (PS2)
Sexy Parodius (PS1)
Zombies vs. Ambulance (PS2)
Non-Violent Games
Marble Mania (Wii)
HM: Tree of Tranquility (Wii)
Ore no Ryouri (PS1)
Tori no Hoshi: Aerial Planet (PS2)
Aquanauts Holiday would also be fun but it's a bit too pricey for me right now.

Animal Main Characters
The Dog Island (Wii)
Finny the Fish & the Seven Waters (PS2)
102 Dalmatians: Puppies to the Rescue (PS1)
Klonoa 2 (PS2)
Hamtaro: Ham Hams Unite (GBC)
Unrealistic Sports Games
Battle Golfer Yui (Genesis)
Pangya: Fantasy Golf (PSP)
Harry Potter Quidditch (PC)
Mario Golf (GBC)
We Ski & Snowboard (Wii)
Sega Superstars Tennis (PS3)
Wii Sports Resort (Wii)
I might play some other games instead, but those are just a few that I'm thinking about right now.