<image>As shown in the figure, given points $$A(0,2)$$, $$B(2,2)$$, $$C(-1,-2)$$, the parabola $$F$$: $$y = x^2 - 2mx + m^2 - 2$$ intersects with the line $$x = -2$$ at point $$P$$.  
$$(1)$$ When the parabola $$F$$ passes through point $$C$$, find its equation;  
$$(2)$$ There are two points $$M(x_1, y_1)$$ and $$N(x_2, y_2)$$ on the parabola $$F$$. If $$-2 \leqslant x_1 < x_2$$, and $$y_1 < y_2$$, determine the value range of $$m$$;  
$$(3)$$ Let the y-coordinate of point $$P$$ be $$y_P$$. Find the minimum value of $$y_P$$. At this time, there are two points $$M(x_1, y_1)$$ and $$N(x_2, y_2)$$ on the parabola $$F$$. If $$x_1 < x_2 \leqslant -2$$, compare the values of $$y_1$$ and $$y_2$$;  
$$(4)$$ When the parabola $$F$$ has common points with line segment $$AB$$, directly write out the value range of $$m$$.