The papers of Zimmer are good references, but they only cover the commutative case (the case of an action on a measure space). For actions of (locally compact) groups on general von Neumann algebras, amenability is defined and studied by Claire Anantharaman-Delaroche in the following papers:
Claire Anantharaman-Delaroche,
Action moyennable d'un groupe localement compact sur une algèbre de von Neumann.(French. English summary)Math. Scand.45(1979), no.2, 289–304. https://www.mscand.dk/article/view/11844
Claire Anantharaman-Delaroche, Action moyennable d'un groupe localement compact sur une algèbre de von Neumann. II.(French. English summary)[Amenable action of a locally compact group on a von Neumann algebra. II] Math. Scand.50(1982), no.2, 251–268.
https://www.mscand.dk/article/view/11958
In the first paper it is already proved that the crossed product $M\bar\rtimes_\alpha\Gamma$ if injective if and only if $M$ is injective and the action $\alpha$ is amenable. Here $\Gamma$ is a discrete group. For general locally compact groups only one direction of this holds.
For actions of (discrete) groups on $C^*$-algebras, amenability is defined and studied in the follow up paper:
Claire Anantharaman-Delaroche, Systèmes dynamiques non commutatifs et moyennabilité.(French)[Noncommutative dynamical systems and amenability] Math. Ann.279(1987), no.2, 297–315. https://link.springer.com/article/10.1007/BF01461725
In this paper it is proved that for an action $\alpha$ of a (discrete) group $\Gamma$ on a $C^*$-algebra $A$, the crossed product $A\rtimes_\alpha \Gamma$ is nuclear if and only if $A$ is nuclear and $\alpha$ is amenable.
Amenable actions of locally compact groups were only defined and studied recently in the preprint (still not published):
https://arxiv.org/abs/2003.03469
Amenability and weak containment for actions of locally compact groups on $C^*$-algebras, by Alcides Buss, Siegfried Echterhoff, Rufus Willett
Further references, and historical background, can be found in that preprint.
You are right. But you already know that in the non-degenerate case the wot/sot closure agrees with the double commutant, which is an algebra.
Even if $A$ is degenerate, unital or not, you can do the following.
Let
$$
p=\inf\{q\in A'':\ q\ \text{ is a projection such that }qa=a\ \text{ for all }a\in A\}.
$$
This infimum exists: the definition above is equivalent to $p$ being the projection onto the subspace $\bigcap_q qH$.
$p\in A''$. Indeed, if $T\in A'$ is selfadjoint, then $TqH=qTH\subset qH$, and so $TpH\subset pH$. This implies that $pTp=Tp$; taking adjoints, $Tp=pT$. If $T\in A'$ it is a linear combination of selfadjoints, so $p\in A''$.
You have $pA=A$, and $(1-p)A=0$, $p\in A''$. Let $A_1=A+\mathbb C\,(1-p)$. It is easy to check that $A_1'=pA'+(1-p)B(H)(1-p)$, and similarly that $A_1''=pA''+ \mathbb C(1-p)$.
You can also check that $\overline{A_1}^{SOT}=\overline{A}^{SOT}+\mathbb C(1-p)$, and that $p\overline{A}^{SOT}=\overline{pA}^{SOT}=\overline{A}^{SOT}_{\vphantom{SOT}}$.
It follows that
$$
\overline{A}^{SOT}=p\overline{A_1}^{SOT}=pA'',
$$
which is an algebra (note that $p\in A'\cap A''$).
The above is most often not necessary, because one considers von Neumann algebras represented non-degenerately (that is, "multiplied by $p$"), and that's probably while it is often glossed over (I'm not even sure if I have seen it explicitly in any textbook).
Best Answer
In a sense you have to distinguish between an abstract and a concrete von Neumann algebra. But, as with C$^*$-algebras, since you can always represent them as concrete, the distinction is not that important.
But it is true that a von Neumann algebra can be represented on a Hilbert space in such a way that it is not equal to its double commutant. For instance you take a II$_1$ factor and consider an irreducible representation: its image will be dense in some (nontrivial) $B(K)$ but of course it cannot be the whole thing. It is hard to imagine that these representations are of any use, so you always represent your vN algebra in a way that suits you best.
So the meaningful question is whether you can tell intrinsically if a C$^*$-algebra is (or, better said, can be represented as) a von Neumann algebra. This is what you would say is an abstract definition of a von Neumann algebra.
Sakai's characterization is in a sense too abstract. Because it is explicitly known what the predual should be: the normal functionals. So a C$^*$-algebra is isomorphic to a concrete von Neumann algebra precisely when the normal functionals separate points.
When people say "weak topology" in the context of von Neumann algebras they are usually referring to the topology induced by the normal functionals, which is the weak$^*$ topology when the algebra is seen as the dual of the normal functionals. In a concrete von Neumann algebra, this is the ultraweak topology; it agrees with the weak operator topology on bounded sets.