ad 1) No, neither implication holds; let $X = \mathbb{R}$. First take $\mu_n = \delta_1$ and $\mu = 0$. Then $(P_2)$ is satisfied, whereas $(P_3)$ is not. Second, take $\mu_n=0$ and $\mu = \delta_2$. Then $(P_3)$ holds, but $(P_2)$ does not.
ad 2) A statement of the equivalences between $(P_1)$, $((P_2)+(P_3))$ and $(P_4)$ can be found in Theorem 3.2 in Resnick, Heavy-Tail Phenomena, Springer 2007. $(P_2)$ should additionally have the criterion that $O$ is relatively compact. Resnick's book is not explicit about the proof; the space $X$ is required to be locally compact and separable, see page 48.
The book can be found here, https://books.google.de/books?id=p8uq2QFw9PUC&lpg=PP1&dq=resnick%202007%20probability&pg=PA52#v=onepage&q=theorem%203.2&f=false
EDIT: another source which I recently encountered: Lindskog, Resnick and Roy, Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps, available at http://projecteuclid.org/euclid.ps/1413896892. In Section 2 a certain type of convergence is the topic and in Theorem 2.1 you might find a portmanteau theorem.
The Helly-Bray theorem also holds for $\mathbb{R}^n$.
"$\Rightarrow$": Assume that $\mu_n \to \mu$ vaguely. By the Portmanteau theorem for vague convergence, $\mu_n(B) \to \mu(B)$ for all bounded $\mu$-continuity Borel sets $B \subseteq \mathbb{R}^n$. For $i = 1, \dots, n$ denote by $D_i \subseteq \mathbb{R}$ the set of continuity points of the marginal measure $\mu_i$ on $\mathbb{R}$. Then $D_i$ is countable and $C := D_1^c \times \dots \times D_n^c$ is dense in $\mathbb{R}^n$. For any point $u \in C$, the set $(-\infty, u]$ is a $\mu$-continuity set. Therefore, $u$ is a continuity point of $F$. Any rectangular box $(a, b]$ with $a, b \in C$ is a $\mu$-continuity set. Any corner $u$ of $(a, b]$ is contained in $C$.
With this in mind, let $x$ be a continuity point of $F$. We can decompose $(-\infty, x]$ into a countable collection of boxes $(a^k, b^k]$ with $a^j, b^j \in C$. Since all these boxes $(a^j, b^j]$ are $\mu$-continuity sets, we get
$F_n(x) = \sum_j \mu_n(a^j, b^j] \to \sum_j \mu(a^j, b^j] = F(x)$ by the bounded convergence theorem.
"$\Leftarrow$": Assume that $F_n(x) \to F(x)$ for all continuity points $x$ of $F$. For a box $(a, b]$ it holds $\mu(a, b] = \Delta^a_b F$ which is an alternating sum over values $F(x)$ with $x$ a corner of $(a, b]$. If $a, b \in C$ then all the corners of $(a, b]$ are contained in $C$ and since $F$ is continuous on $C$ we get $\mu_n(a, b] = \Delta^a_b F_n \to \Delta^a_b F = \mu(a, b]$. Let $g : \mathbb{R}^n \to \mathbb{R}$ be continuous with compact support. Then $\textrm{supp}(g) \subseteq (a, b]$ for some $a, b \in C$. Let $\varepsilon > 0$. Since $g$ is uniformly continuous on $(a, b]$ and $C$ is dense in $\mathbb{R}^n$ we can partition $(a, b]$ into finitely many boxes $(a^j, b^j]$, $j = 1, \dots, m$ with $a^j, b^j \in C$ such that $\sup_{x \in (a^j, b^j]} |g(x) - g(b^j)| < \varepsilon$ for all $j$. Decompose $\int g d\mu = \sum_j \int_{(a^j, b^j]} g d\mu$. We can approximate
$$\left|\int g d\mu - \sum_j g(b^j) \mu(a^j, b^j]\right| = \left|\sum_j \int_{(a^j, b^j]} (g(x) - g(b^j)) \mu(dx)\right| \\
\leq \sum_j \sup_{x \in (a^j, b^j]} |g(x) - g(b^j)| \mu(a^j, b^j] < \varepsilon \cdot \mu(a, b]$$
and similarly for all the $\mu_n$. It follows
$$\left|\int g d\mu_n - \int g d\mu\right| \leq \left| \int g d\mu_n - \sum_j g(b^j) \mu_n(a^j, b^j]\right| + \left| \sum_j g(b^j)(\mu_n(a^j, b^j] - \mu(a^j, b^j])\right| \\
+ \left| \int g d\mu - \sum_j g(b^j) \mu(a^j, b^j] \right| \leq 2\varepsilon + \lVert g \rVert \sum_j |\mu_n(a^j, b^j] - \mu(a^j, b^j]|.$$
As $n \to \infty$, the right-hand side converges to $0$ (the sum is finite) and we get
$\limsup_n |\int g d\mu_n - \int g d\mu| \leq 2 \varepsilon$. Since this is true for all $\varepsilon$, $\int g d\mu_n \to \int g d\mu$. Therefore, $\mu_n \to \mu$ vaguely.
Best Answer
Here is the general notion of vague convergence, taken from Olav Kallenberg's Foundations of Modern Probability (2nd edition).
Let $S$ be a locally compact, second countable Hausdorff space equipped with its Borel $\sigma$-field ${\cal S}$. Let $\hat{\cal S}$ be the ring of relatively compact Borel sets, and ${\cal M}(S)$ the space of locally finite non-negative measures. Locally finite means that $\mu(B)<\infty$ when $B\in \hat{\cal S}$.
The vague topology on ${\cal M}(S)$ is the topology generated by the mappings $\mu\mapsto \int f\ d\mu$ for every $f$ a non-negative continuous function with compact support.