Skorokhod's theorem and summary of convergence of sequence of RVs This question is about convergence of RVs (when convergence in one sense implies other convergence modes). 
I would like to have a big picture on convergence modes and various implications between them. My sources are books, older posts here, and Wikipedia, and I hope to obtain a big beautiful picture with your help. I will update this post with your replies and comments (I hope this is in line with MSE guidelines).
Start with the big picture (main source: this). I write $X_n\to X$ for RVs and $F_n\to F$ for distributions (laws).
\begin{matrix}
X_n \xrightarrow{L^s} X \implies X_n \xrightarrow{L^{s-1}} X \implies \cdots \implies 
& X_n \xrightarrow{L^1}X & \\
& & \hspace{-25mm}\searrow \nwarrow_{**} \\
& & X_n \xrightarrow{p} X \implies F_n \to F \\
& \Uparrow_{*} & \hspace{-25mm}\nearrow & \\
& X_n \xrightarrow{a.s.} X &
\end{matrix}
(The picture is not so typographically beautiful actually! I don't know how to rotate implication symbols in MathJax. So rotated single arrows have to be intended as implications.)
A.s. convergence implies convergence in $L^1$ ($\Uparrow_{*}$) under the hypotesis: $\exists Y: |X_n|<Y, \mathbb{E}(Y)<\infty$.
Convergence in p. implies convergence in $L^1$ ($\nwarrow_{**}$) under the hypothesis of  uniformly integrable (u.i.) $X_n$, i.e.
$$ \lim_{a\to\infty} \sup_n \mathbb{E}(|X_n| \mathbb{1}_{\{|X_n|>a\}})=0.$$ A sufficient (simpler) condition which implies u.i. is $\exists \epsilon>0: \sup_n \mathbb{E}(|X_n|^{1+\epsilon})<\infty.$
I would like to complete the picture, i.e. know if there exist conditions which ensure  stronger convergence modes from weaker ones (for example, when convergence in p. implies convergence a.s.) or, more in general, conditions under which a convergence mode implies others (for example when convergence in the first mean implies convergence a.s.).
** Big picture (after the answer) ** 

Implication with one asterisk: valid for a subsequence.
Implication with two asterisks: under the uniform integrability condition (see also the characterizations proposed above and in the answer).
Implication with three asterisks: under the further condition that $X$ is a.s. constant (i.e. the distribution $F$ is a point mass).
 A: 
I would like to complete the picture...

Here we go.
First, the implication from almost sure convergence to convergence in $L^1$ under condition $\ast$ is weaker than the route from almost sure convergence to convergence in probability to convergence in $L^1$ under condition $\ast\ast$, since $\ast$ implies $\ast\ast$. Thus, implication $\ast$ should probably be omitted altogether.
Second, the convergence $F_n\to F$ should be renamed convergence in distribution. Recall that convergence in distribution is equivalent to pointwise convergence $F_n(x)\to F(x)$ for every $x$ where $F$ is continuous, not to the pointwise convergence $F_n\to F$.
Third, convergence in distribution implies convergence in probability under the condition that $X$ is almost surely constant.
Fourth, convergence in probability implies convergence almost sure of a subsequence.
Finally, still another characterization of the uniform integrability of a family $\mathcal X$ of random variables is the existence of some nonnegative function $\varphi$ on $[0,\infty)$ such that $\varphi(x)/x\to\infty$ when $x\to\infty$ and $E(\varphi(|X|))\leqslant1$ for every $X$ in $\mathcal X$. This includes the case $\varphi:x\mapsto x^{1+\varepsilon}$ mentioned in your post but also cases such as $\varphi:x\mapsto x(\log x)^+$.
