what is the joint distribution of these two random variables? what is the joint distribution of two random variables,$∑_iX_iY_i$ and $(∑_iX_i)(∑_jY_j)$?
Note that since $n$ is a large number and all the random variables are $iid$, using central limit theorem, we can say that $∑_iX_iY_i$, $(∑_iX_i)$ and $(∑_jY_j)$ are approximately normal random variables and $(∑_iX_i)$$(∑_jY_j)$ is the product of two normal random variables which would have Normal Product Distribution.
 A: First of all, if $X_i$ are iid then $\sum X_i$ is not approximately normal, check carefully the statement of the CLT, so your statements about the limiting case are not correct. 
Secondly, if $X_i$ are not independent, you have to know their joint distribution to know the dependence structure between them. Knowing the joint distribution of a random vector $(X_i)_{i=1}^n$, you can easily recover the distribution of a function of this vector
$$
f(X_1,\dots,X_n) := \sum X_i.
$$
In case $X_i$ are independent, you have
$$
  \mathsf P(X_1+X_2\leq t) = \int_\Bbb R F_2(t-x_1)\mathrm dF_1(x_1)
$$
where $F_i$ is a CDF of $X_i$. I hope, you know how to extend this formula to the case of more than $2$ variable by induction.
Finally, when you need to deal with products you have 
$$
  \mathsf P(XY\leq t) = \int_\Bbb R \mathsf P(Xy\leq t)\mathrm dF_Y(y) 
$$
$$
= \int_0^\infty F_X(t/y)\mathrm dF_y(y)+\int_{-\infty}^0 (1-F_X(t/y))\mathrm dF_Y(y) + 1\{t\geq 0\}\cdot\mathsf P(Y = 0).
$$
I think, you now have all ingridients.
