This not an answer, but a comment (I am not entitled). For any $F$ which satisfies conditions to be described below there are precisely two quadrilaterals  $ABCD$  which have area $A$ and side lengths $a,b,c,d$.  To be precise, I am using directed areas (i.e. $A\wedge B+B\wedge C+C\wedge D+D\wedge A$) and side lengths are $|AB|=a$ and so on cyclically. Hence the required max and min are the largest and smallest values for $F$ which satisfy the given condition.  

The condition on $F$ is the positivity of  an explicit  sextic polynomial whose coefficients are functions of the side lengths.  Hence the required optimal values are roots of this polynomials.  The polynomilal is, in fact, a cubic in $F^2$ so that it can be solved by radicals.  The concrete form of the cubic is too complicated to quote here.