Can a plane reflection in 3-space be written as the product of 3 line reflections? 

For any reflection (through a line or plane), we can make the line or plane pass through the origin by a translation, then do the reflection, and finally undo the translation. So if the reflection is given by matrix $A$, then the point $x$ is first moved to $x-b$ for some fixed point $b$ on the line or plane, then we do the reflection to get $A(x-b)=Ax-Ab$, and finally translate back to get $Ax-Ab+b$. So the overall transformation can be written as $Ax+a$ for some $a$. 

Now let the three line reflections be given by matrices $A,B,C$ and vectors $a,b,c$, and the plane reflection be $Dx+d$. Suppose to the contrary that the composition of the three line reflections is the plane reflection: $$ Dx+d=C(B(Ax+a)+b)+c=CBAx+CBa+Cb+c\\ \implies Dx=CBAx+v$$for some fixed vector $v$. Since this equality holds for all $x$, upon differentiation we get $$ D=CBA$$However, this equality cannot hold since $\det D=-1$ while $\det CBA=1$. Thus the composition of three line reflections cannot be a plane reflection.