Throughout this answer $(M,g)$ will be an asymptotically flat spacetime. Future and past null infinity $\cal{I}^\pm$ are topologically ${\cal I}^\pm \simeq \mathbb{R}\times S^2$. This $S^2$ part is called the celestial sphere. I shall also, for definiteness, focus on $\cal I^+$.
We shall also for simplicity employ the BMS approach of working in terms of falloff conditions inside the physical spacetime. This is meant also to align this answer with current literature on the topic, specifically the book "Lectures on the IR Structure of Gravity and Gauge Theories". For a coordinate-free review employing the rigorous construction which realizes null infinity as a boundary in one unphysical spacetime, see e.g. this review by Ashtekar.
In any spacetime one may find, locally, coordinates $(u,r,\Theta^A)$ adapted to null surfaces, on which the metric tensor may be written as $$ds^2=-Udu^2-2e^{2\beta}dudr+g_{AB}\left(d\Theta^A+\frac{1}{2}U^Adu\right)\left(d\Theta^B+\frac{1}{2}U^Bdu\right)\tag{5.1.2},$$ where $\Theta^A$ coordinates on a $2$-sphere and with the condition that $$\partial_r\left(\frac{g_{AB}}{r^2}\right)=0.\tag{5.1.3}$$ This choice of coordinates is called Bondi gauge.
Asymptotic flatness near $\cal{I}^+$ will be taken to mean that it is possible to find such a coordinate system $(u,r,\Theta^A)$ where we have the ranges $u\in \mathbb{R}$, $r\in [r_0,+\infty)$ and $\Theta^A$ parameterize the whole sphere, and where the metric tensor components satisfy a specific falloff condition I'll get to in a moment.
From now on I shall assume that the coordinates on $S^2$ are holomorphic coordinates. This is just a fancy way of saying that you take the usual stereographic projection of $S^2$ onto $\mathbb{R}^2$, identifies $\mathbb{R}^2\simeq \mathbb{C}$ and then trades the coordinates $(x,y)$ of $\mathbb{R}^2$ by $z = x+iy$ and $\bar{z} = x-iy$.
The falloff behavior of the metric tensor components is then required to be: $$g_{uu}=-1+{\cal O}\left(\frac{1}{r}\right),\quad g_{ur}=-1+{\cal O}\left(\frac{1}{r^2}\right),\quad g_{uz}={\cal O}(1),\\ g_{zz}={\cal O}(r),\quad g_{z\bar{z}}=r^2\gamma_{z\bar{z}}+{\cal O}(1),\quad g_{rr}=g_{rz}=0.\tag{5.1.5}$$
In this approach one reaches ${\cal I}^+$ asymptotically by holding $(u,z,\bar{z})$ fixed and taking $r\to \infty$.
In this approach the BMS group arises when one looks for the diffeomorphisms which preserve both the Bondi gauge (5.1.2), (5.1.3) and the falloff conditions (5.1.5). This group may be shown to be a semi-direct product of the so-called supertranslations with the universal cover of the Lorentz group, $SL(2,\mathbb{C})$. The usual analysis is to describe such diffeomorphisms by their generators and require the resulting finite transformations to be globally defined.
The supertranslations end up being parameterized by functions $f(z,\bar{z})$ on the sphere and in coordinates they act by taking $u\mapsto u+f(z,\bar{z})$. They are angle-dependent translations along the generators of $\mathcal{I}^+$.
The Lorentz transformations appear then to be generated by the global conformal Killing vectors of the celestial sphere and the $SL(2,\mathbb{C})$ part of the BMS group acts as the global conformal group of the celestial sphere.
The superrotations are not included in that discussion. The reason is exactly because in defining the standard BMS group one expects globally defined generators. Superrotations arise when you allow the full set of conformal Killing vector fields of the celestial sphere. In other words you allow for meromorphic vector fields which might not be globally defined nor give rise to global conformal transformations.
Now notice how the definition is framed: superrotations means one infinite dimensional generalization of the Lorentz transformations by allowing the full locally defined conformal Killing vectors. So despite the name this generalizes the whole Lorentz group, not just the rotations. For more on superrotations read the section 5.3 of "Lectures on the IR" book or this thesis focused on superrotations.
So answering your questions :
No, the standard BMS group is the semidirect product of supertranslations and the universal cover of the Lorentz group $SL(2,\mathbb{C})$ allowing only globally defined conformal transformations on the celestial sphere;
Superrotations are to be interpreted as further allowing the locally defined conformal Killing vector fields. That the BMS group is an asymptotic symmetry has been proved by its equivalence to Weinberg's soft graviton theorem (c.f. the "Lectures on the IR" books and references therein, also see this paper). That superrotations also give rise to asymptotic symmetries has been proved to tree level and this apparently has already received one-loop corrections. As far as I know there is no argument that this is the whole of asymptotic symmetries, in the sense that what has been said so far is that these are a subset of such symmetries.
