How can I save both the plot and the legend together?

This is how to save the graph, legend and all:

Another way would be to use Rasterize:

Rasterize[Plot[...]]

The legend and the graph is now one image.

You can define :

sol = Solve[Expr1 == Expr2, p][[1, 1]] ;
plot = Plot[{Expr1, Expr2, Expr3}, {p, 0, 1}, 
        BaseStyle -> AbsoluteThickness[4], 
        PlotLegends -> Placed[LineLegend["Expressions", BaseStyle -> AbsoluteThickness[4]], {Left, Center}], 
        Epilog -> Inset[sol, {0.5, 0.5}]]

and export with

Export["/tmp/plot.png", plot]

Or you can use the full form position specification with Placed to position legend "inside" the graph and use Legended to put any additional legends.

Legended[Plot[{Expr1, Expr2, Expr3}, {p, 0, 1}, 
  BaseStyle -> AbsoluteThickness[4], 
  PlotLegends -> 
   Placed[LineLegend["Expressions", 
     BaseStyle -> AbsoluteThickness[4]], {1.05, 0.5}]], 
   Placed[First@First@Solve[Expr1 == Expr2, p], {0.5, 1}]]