Short answers, pulled from the story.
Madhouse, Inc. was founded in 1972 by a group of animators including Masao Maruyama, Osamu Dezaki, and Yoshiaki Kawajiri. These founders left Mushi Production to establish a new studio with funding from Yutaka Fujioka of Tokyo Movie.
The first project produced by Madhouse, Inc. was Aim for the Ace! in 1973. This co-production with Tokyo Movie showcased a fresh, fluid animation style distinct from the stiff movements of earlier works.
Notable directors associated with Madhouse, Inc. include Satoshi Kon and Yoshiaki Kawajiri. Kawajiri directed action films such as Wicked City, Demon City Shinjuku, and Ninja Scroll, while Kon created psychological masterpieces like Perfect Blue, Millennium Actress, Tokyo Godfathers, and Paprika.
Yes, Madhouse, Inc. collaborated with Studio Ghibli to provide key animation assistance for films such as My Neighbor Totoro, Spirited Away, and Howl's Moving Castle. These partnerships demonstrated mutual respect and allowed Madhouse to learn from some of the greatest animators in the world.
In the 2000s, Madhouse, Inc. collaborated with Western companies to produce content such as Marvel Entertainment adaptations of Blade, Iron Man, Wolverine, and X-Men, and an anthology adaptation of Batman for DC Entertainment. The studio also produced Todd McFarlane's Spawn for HBO and The Animatrix, an anthology adaptation of the Matrix franchise.
Critically acclaimed series produced by Madhouse, Inc. during the 2010s and 2020s include Death Note, Black Lagoon, and Hunter × Hunter. The studio also released successful films such as Wolf Children, Summer Wars, and Redline during this period.