Solution: (1), obtained from c (0 0,4), -4a=4, so a=- 1. Substitute a (- 1, 0) into y =-x 2+bx+4 to get b=3. That is, the analytical formula of parabola is: y =-x 2+3x+4.

(2) Substitute point D(m, m+ 1) into the analytical formula of parabola to get: m+ 1 =-m 2+3m+4 to get m=3(m=- 1), that is, point D (3,4).

The analytical formula of another straight line BC is: y=-x+4. If the intersection point D is perpendicular to the straight line BC, its analytical formula is y=x+ 1, and the coordinate of the point D is (0, 1).

(3) If ∠ DBP = 45, the point P must be located at the lower side of the straight line BC, the intersection point D is de perpendicular to BC, and the intersection point P is PF perpendicular to F. By using the triangle BDE similar to the triangle BPF, the ratio of PF to BP can be found to be 3: 5, PF can be 3k, and PB can be 4-5k, and then k=22/25 can be obtained by substituting it into the analytical formula of parabola.

P(-2/5，66/25)。 Note: The length of DE can be obtained by equal area method.