Tikz: Draw to the bounding box

Following my previous question (when I asked that one I thought the rest would be relatively easier):

\documentclass[tikz, border=1cm]{standalone}
\begin{document}
\begin{tikzpicture}
[plotmark/.style = {%
solid,
fill=red,
circle,
inner sep = 0pt,
minimum size = 4pt}]

\begin{scope}[rotate=-{30},local bounding box=a]
\draw (0,0) |- (4,4) |- (6,2) |- (4,0) |- (2,-2) |- cycle;
\draw (4, 0) -| (2,4)
(0,2) -| (4,0);
\coordinate (A) at (2,-2);
\coordinate (B) at (2,4);
\node[plotmark] at (A){};
\node[plotmark] at (B){};
\end{scope}
\draw (a.south west) rectangle (a.north east);
\end{tikzpicture}
\end{document}


1. What are my options to fill the bounding box but exclude the inner shape? (Torbjørn T. provided an answer with on background layer but advised to ask it as a separate question.) How can I do it with clip and scope?

2. How can I draw a line which passes through the two red points and intersects with the bounding box?

UPDATE: Without backgrounds and clip.

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{intersections,calc}
\begin{document}
\begin{tikzpicture}
[plotmark/.style = {%
solid,
fill=red,
circle,
inner sep = 0pt,
minimum size = 4pt}]

\begin{scope}[rotate=-{30},local bounding box=a]
\draw[fill=gray,even odd rule] (0,0) rectangle ++(2,2) (2,0) rectangle ++(2,2) (0,-2) rectangle ++(2,2) (2,-2) rectangle ++(2,2) (4,-2) rectangle ++(2,2) (2,-4) rectangle ++(2,2)
(a.north west) -- (a.north east) -- (a.south east) -- (a.south west)
--(a.north west);
\coordinate (A) at (2,-4);
\coordinate (B) at (2,2);
\node[plotmark] at (A){};
\node[plotmark] at (B){};
\end{scope}
\draw[name path global=bbox] (a.south west) rectangle
(a.north east);
\path[overlay,name path=line] let \p1=(A),\p2=(B),\p3=(a.south),\p4=(a.north),\n1={((\y4-\y3)/(\y2-\y1))} in
($(A)-\n1*(\x2-\x1,\y2-\y1)$)
--
($(B)+\n1*(\x2-\x1,\y2-\y1)$);
\draw[name intersections={of=bbox and line},fill=blue]
(intersection-1) node[plotmark]{} -- (intersection-2) node[plotmark]{};
\end{tikzpicture}
\end{document}


\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{backgrounds,intersections,calc}
\begin{document}
\begin{tikzpicture}
[plotmark/.style = {%
solid,
fill=red,
circle,
inner sep = 0pt,
minimum size = 4pt}]

\begin{scope}[rotate=-{30},local bounding box=a]
\draw[fill=white] (0,0) rectangle ++(2,2) (2,0) rectangle ++(2,2) (0,-2) rectangle ++(2,2) (2,-2) rectangle ++(2,2) (4,-2) rectangle ++(2,2) (2,-4) rectangle ++(2,2);
\coordinate (A) at (2,-4);
\coordinate (B) at (2,2);
\node[plotmark] at (A){};
\node[plotmark] at (B){};
\end{scope}
\begin{scope}[on background layer]
\draw[fill=gray,name path global=bbox] (current bounding box.south west) rectangle
(current bounding box.north east);
\end{scope}
\path[overlay,name path=line] let \p1=(A),\p2=(B),\p3=(a.south),\p4=(a.north),\n1={((\y4-\y3)/(\y2-\y1))} in
($(A)-\n1*(\x2-\x1,\y2-\y1)$)
--
($(B)+\n1*(\x2-\x1,\y2-\y1)$);
\draw[name intersections={of=bbox and line},fill=blue]
(intersection-1) circle (2pt) -- (intersection-2) circle (2pt);
\end{tikzpicture}
\end{document}


There are multiple ways, one of them. The filled area and the extra lines to the bounding box are drawn on the background layer with clipping.

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{backgrounds}
\begin{document}
\begin{tikzpicture}[
plotmark/.style = {%
solid,
fill=red,
circle,
inner sep=0pt,
minimum size=4pt,
},
]
\begin{scope}[rotate=-30,local bounding box=bb]
\draw
(2, 4)
-- (2, -2) coordinate (b)
-- (4, -2) coordinate (c)
-- (4, 4) coordinate (f)
-- (0, 4) coordinate (g)
-- (0, 0) coordinate (a)
-- (6, 0) coordinate (d)
-- (6, 2) coordinate (e)
-- (0, 2)
;
\coordinate (A) at (2, -2);
\coordinate (B) at (2, 4);
\node[plotmark] at (A){};
\node[plotmark] at (B){};
\end{scope}

\begin{pgfonlayer}{background}
\clip
(bb.south west) rectangle (bb.north east)
{[rotate=-30]
(a) -| (b) -- (c) |- (d) -- (e) -| (f) -- (g) |- cycle
}
;
\fill[lightgray]
(bb.south west) rectangle (bb.north east)
;
\draw[rotate=-30]
(A) -- ++(0, -2)
(B) -- ++(0, 2)
;
\end{pgfonlayer}

\draw (bb.south west) rectangle (bb.north east);
\end{tikzpicture}
\end{document}