For the first question,
"So, let us assume X and Y are not independent. How can one obtain regularity of the joint density having only properties from the marginal densities? Is it possible? Is there any criteria?",
Perhaps Copula could help you.
lets start with a example. In this example I want to generate distribution
$F_{(X,Y)}(x,y)$ such that their marginals be $F_1$ and $F_2$ and
$F_{(X,Y)}(t_1,t_2) \neq F_{1}(t_1) F_{2}(t_2) $ (so they are
not independent).
lets $ F_1$ and $ F_2$ are continues and known.
let $\Phi$ is cumulative distribution of uni-variate standard normal . suppose $$(e_1,e_2) \sim \Phi_2$$ where $\Phi_2$ is CDF of standard bi-variate normal distribution, that is,
\begin{eqnarray}
(e_1,e_2)\sim Normal \bigg( \left(
\begin{array}{c}
0 \\
0
\end{array}
\right) ,
\left(
\begin{array}{cc}
1 & \rho \\
\rho & 1
\end{array}
\right)
\bigg)
\end{eqnarray}
define
$$X=F^{-1}_1(\Phi(e_1))$$
$$Y=F^{-1}_2(\Phi(e_2))$$
so
$$F_X(t_1)=P(X\leq t_1)=P(F^{-1}_1(\Phi(e_1))\leq t_1)$$
$$=P\bigg(e_1 \leq \Phi^{-1}\left( F_1( t_1) \right) \bigg)=
\Phi \bigg( \Phi^{-1}\left( F_1( t_1) \right) \bigg)=F_1(t_1)$$
so $X\sim F_1$. similarity $Y\sim F_2$. In next step I show they are not independent.
$$F_{(X,Y)}(t_1,t_2)=P(X\leq t_1 , Y\leq t_2)$$
$$=P(F^{-1}_1(\Phi(e_1))\leq t_1 , F^{-1}_2(\Phi(e_2))\leq t_2)$$
$$=P\bigg(e_1 \leq \Phi^{-1}\left( F_1( t_1) \right)
, e_2 \leq \Phi^{-1}\left( F_2( t_2) \right) \bigg)$$
$$=\Phi_2\bigg(\Phi^{-1}\left( F_1( t_1) \right)
, \Phi^{-1}\left( F_2( t_2) \right) \bigg)$$
so
$$F_{(X,Y)}(t_1,t_2)=\Phi_2\bigg(\Phi^{-1}\left( F_1( t_1) \right)
, \Phi^{-1}\left( F_2( t_2) \right) \bigg)$$
now if $\rho \neq 0 $
$$F_{(X,Y)}(t_1,t_2)=\Phi_2\bigg(\Phi^{-1}\left( F_1( t_1) \right)
, \Phi^{-1}\left( F_2( t_2) \right) \bigg)$$
$$\neq F_1( t_1) \times F_2( t_2) $$
but if $\rho = 0 $
$$F_{(X,Y)}(t_1,t_2)=\Phi_2\bigg(\Phi^{-1}\left( F_1( t_1) \right)
, \Phi^{-1}\left( F_2( t_2) \right) \bigg)$$
$$=\Phi \bigg(\Phi^{-1}\left( F_1( t_1) \right) \bigg)
\times \Phi \bigg(\Phi^{-1}\left( F_2( t_2) \right) \bigg)$$
$$= F_1( t_1) \times F_2( t_2) $$
This method known as "Copula".
In general case let $F_1$ and $F_2$ are known.
$$(X,Y)\sim F_{(X,Y)}(t_1,t_2)=c\left(F_1(t_1),F_2(t_2)\right)$$
Now If $$c(a,b)=a\times b$$ so $X$ and $Y$ are independent. if $$c(a,b)\neq a\times b$$ they are dependent. you can have a different way by choosing
different $c(a,b)$ . In this paper you can see three different choise for $c\left(F_1(t_1),F_2(t_2)\right)$.
For the second question, find example with such condition ,you may find a $c\left(F_1(t_1),F_2(t_2)\right)$ such that satisfy your condition.