Can you catch a skin disorder from a cat? or bacteria ?

Yes, yes: Mange and some other fungal infections can be transmitted to humans but this is rare. If a cat bites or scratches you deep enough, you can get a bacterial infection. One of these bacteria is pasturella. There is also a condition called cat scratch fever that is an infection transmitted from cats to people. Finally, there are some amebic parasitic infections that can be transmitted.

Agree w Dr. Bansal: Pasturella & Cat scratch disease could be transmitted that way. Other infections that people can get from cats include: Campylobacter, Ringworm, Roundworm, Hookworm, Tapeworm, Rabies, Fleas (very temporarily), Toxoplasmosis, Giardiasis, Cryptosporidiosis & Salmonellosis.