$\def\d{\mathrm{d}}\def\dto{\stackrel{\mathrm{d}}{→}}\def\aeto{\xrightarrow{\mathrm{a.e}}}\def\stdom{\geqslant_{\mathrm{st}}}\def\peq{\mathrel{\phantom{=}}{}}$Even if $\{X_n\}$, $\{Y_n\}$, $\{W_n\}$ may be defined on different probability spaces, a product probability space can be constructed to embed all these random variables since only their distributions are involved in the propositions to be proved. Hence it can be assumed without loss of generality that all the random variables are in the same probability space.
For question 1, your proof of $E(|W|) < +∞$ is correct. Note that $X_n \dto X$, $Y_n \dto Y$, $W_n \dto W$ imply that $F_{X_n} \aeto F_X$, $F_{Y_n} \aeto F_Y$,
$F_{W_n} \aeto F_W$ since $F_X$, $F_Y$, $F_W$ have at most countably many points of jump. Because $X_n \stdom W_n$, so $F_{X_n} \leqslant F_{W_n}$ and\begin{align*}
&\peq E(X_n) - E(W_n)\\
&= {\small \left( \int_0^{+∞} (1 - F_{X_n}(x)) \,\d x - \int_{-∞}^0 F_{X_n}(x) \,\d x \right) - \left( \int_0^{+∞} (1 - F_{W_n}(x)) \,\d x - \int_{-∞}^0 F_{W_n}(x) \,\d x \right)}\\
&= \int_{-∞}^{+∞} (F_{W_n}(x) - F_{X_n}(x)) \,\d x.
\end{align*}
Since $X_n \dto X$ and $W_n \dto W$, then $X \stdom W$ and analogously,$$
E(X) - E(W) = \int_{-∞}^{+∞} (F_W(x) - F_X(x)) \,\d x.
$$
By Fatou's lemma,\begin{gather*}
\varlimsup_{n → ∞} (E(X_n) - E(W_n)) = \varlimsup_{n → ∞} \int_{-∞}^{+∞} (F_{W_n}(x) - F_{X_n}(x)) \,\d x\\
\leqslant \int_0^{+∞} \lim_{n → ∞} (F_{W_n}(x) - F_{X_n}(x)) \,\d x = \int_0^{+∞} (F_W(x) - F_X(x)) \,\d x = E(X) - E(W),
\end{gather*}
combining with $\lim\limits_{n → ∞} E(X_n) = E(X)$ yields $\varliminf\limits_{n → ∞} E(W_n) \geqslant E(W)$.
Because $W_n \stdom Y_n$, so $F_{W_n} \leqslant F_{Y_n}$ and analogously,$$
\varlimsup_{n → ∞} (E(W_n) - E(Y_n)) \leqslant E(W) - E(Y),
$$
combining with $\lim\limits_{n → ∞} E(Y_n) = E(Y)$ yields $\varlimsup\limits_{n → ∞} E(W_n) \leqslant E(W)$. Thus $\lim\limits_{n → ∞} E(W_n) = E(W)$.
If $F_{X_n} \leqslant F_{W_n} \leqslant F_{Y_n}$ is only known to be true on a dense set $D$, it can be proved that $F_{X_n}(x) \leqslant F_{W_n}(x) \leqslant F_{Y_n}(x)$ holds for all $x$ where $F_{X_n}$, $F_{W_n}$, $F_{Y_n}$ are continuous since for any $x \in \mathbb{R}$, there exists $\{x_n\} \subseteq D$ such that $x_n \searrow x$ when $n → ∞$ (otherwise there exists $δ > 0$ such that $D \cap (x, x + δ) = \varnothing$). Thus $X_n \stdom W_n \stdom Y_n$ is still true.
For question 2, since $X_n \stdom W_n$ implies that $X_n = X_n^+ \stdom W_n^+ \stdom 0$, so replacing $X_n$, $Y_n$, $W_n$ in question 1 by $X_n^+$, $0$, $W_n^+$ yields $\lim\limits_{n → ∞} E(W_n^+) = E(W^+)$, and analogously $\lim\limits_{n → ∞} E(W_n^-) = E(W^-)$, then$$
\lim_{n → ∞} E(|W_n|) = \lim_{n → ∞} E(W_n^+) + \lim_{n → ∞} E(W_n^-) = E(W^+) + E(W^-) = E(W).$$
Best Answer
Let $(x,y,w) \in \mathbb{R}^3,$
$$|\varphi_{(X_n,u_n,y_n)}(x,y,w)-\varphi_{(X,u_0,y_0)}(x,y,w)|=|e^{i(u_ny+wy_n)}\varphi_{X_n}(x)-e^{i(u_0y+wy_0)}\varphi_X(x)|\leq|e^{i(u_ny+y_nw)}(\varphi_{X_n}(x)-\varphi_{X}(x))|+|\varphi_{X}(x)(e^{i(u_ny+y_nw)}-e^{i(u_0y+wy_0)})|$$
and the limit is $0,$ we conclude with the continuous mapping theorem