I agree, your plant is
Dwarf Umbrella Tree (Heptapleurum arboricola) From what I can see with my old eyes, the white stuff does look like fungus which can be caused by heavy, moisture retaining soil and/or over-watering. Your plant would probably enjoy being outdoors in the fresh air (partial shade location) during the warmer months of the year.
I'm wondering if that container might be a bit large for the size of the tree; depending on the size of the root mass, it may be able to grow in a smaller pot. Those side branches growing off the main trunk could be lopped off and rooted for new plants. If the main trunk is viable, new shoots should grow once the branches are cut. In the second photo, I see a short branch with no leaves, I'd cut that back to see if there is any live wood and if it is alive, new branches should sprout, giving the shrub a more uniform shape.
I see a beautiful kitty in your photo and hopefully he/she isn't like a cat we had at one time who we had to supervise closely because she liked to eat my house plants. Schefflera are toxic to cats and dogs:
http://www.aspca.org/pet-care/...
