Are you familioar with the identity for the tangent of added angles?
Call the small angle x, and the unknown length CD = L. Then:
1/L = tan x
4/L = tan(x+35)
Hence: tan(x+35) = 4 tan x
Now use the formula for tan(x+35) to get:
Now solve for . Almost done: You know that 1/L = tan x, so solve for L.
As for the line you drew from C that intersects with the hypotenuse - no, it is not a bisector unless the right triangle is isocoles. Consider that a bisector of a right angle is 45 degees, and the perpendicular intersection of the hypotenuse is 90 degrees. Since angles of a triangle add to 180, the angle A would have to be 45 degrees for this to work.