I think that the following question has a positive answer. Yet, I haven't managed to find it.
Consider the structure $\{0,1\}$ with operations $\lor,\land,\lnot$ defined in a usual way.
Is it true that the theory of this structure is axiomatizable by the axioms of Boolean algebra (written without $\forall$):
Сan you please recommend me some related basic level books (on logic or model theory), where I can find the proof.
P.S. I would be grateful if you told me the standard name of the structure.
It depends exactly what you mean by theory in this context.
One crucial observation is that each of the sentences you've written down is an equation - that is, the universal closure of an expression of the form "$t=s$." Now equations are preserved by Cartesian products, so for example every equation which holds in the two-element Boolean algebra ${\bf 2}$ also holds in the four-element Boolean algebra ${\bf 4}$. Consequently the sentences you've written down do not entail (for example) the sentence $$\forall x,y,z(x=y\vee y=z\vee z=x),$$ which is true in ${\bf 2}$ but false in ${\bf 4}$. Another example of such a sentence would be $$\forall x(x=0\vee x=1).$$
(I'm not sure that this "bolded number" notation is standard, but it is fairly common. In general, when it's clear from context that we're talking about structures in some class $\mathcal{K}$ where there is at most one structure up to isomorphism of each finite size, the notation "${\bf n}$" for $n\in\mathbb{N}$ is often used to refer to the isomorphism type of that unique structure. E.g. if we're talking about linear orderings, you may see "${\bf 2}$" used to denote the linear order with two elements, and so forth. I've also seen things like "$\mathbb{B}_2$" used to denote the unique-up-to-isomorphism Boolean algebra with two elements.)
This means that if we read "theory" as "first-order theory," your question has a negative answer. However, if we look at weaker logics than first-order logic, things change substantially: in fact that set of equations does generate the entire equational theory of ${\bf 2}$, that is, the set of all equations true in ${\bf 2}$. Equational logic is a very small fragment of first-order logic, and is studied in universal algebra.
I recommend Bergman's freely-available-online book as an introduction to universal algebra and equational logic; for (first-order) model theory, see the discussion here.
While not directly relevant to your question, let me as a coda present the first serious theorem in equational logic/universal algebra. Above I mentioned that Cartesian products preserve equations; it's easy to show that so do homomorphisms and substructures. These three facts provide useful tools for establishing negative results about expressibility/axiomatizability in equational logic.
However, we may now ask for a positive result: given an algebra $\mathcal{A}$, what is the set of all algebras satisfying the full equational theory of $\mathcal{A}$? This is called the variety generated by $\mathcal{A}$, and denoted "$\mathsf{Var}(\mathcal{A})$." The above paragraph gives a lower bound on $\mathsf{Var}(\mathcal{A})$: letting $\mathbb{H}$,$\mathbb{S}$,$\mathbb{P}$ be the operations on classes of structures defined by
$\mathbb{H}(\mathfrak{X})$ = the set of all homomorphic images of algebras in $\mathfrak{X}$,
$\mathbb{S}(\mathfrak{X})$ = the set of all substructures of algebras in $\mathfrak{X}$, and
$\mathbb{P}(\mathfrak{X})$ = the set of all Cartesian products of algebras in $\mathfrak{X}$,
we have $\mathsf{Var}(\mathcal{A})\supseteq\mathbb{H}(\mathbb{S}(\mathbb{P}(\{\mathcal{A}\})))$ (or "$\mathsf{Var}(\mathcal{A})\supseteq\mathbb{HSP}(\mathcal{A})$" adopting minor abuse of notation).
It turns out that this is sharp! This is Birkhoff's HSP theorem; there are several questions on this site discussing it, and I've sketched a proof here.
(As a coda to my coda, there is a similar result for first-order logic, but the algebraic operations involved are more complicated: the class of structures with the same first-order theory as a given structure $\mathcal{A}$ is the smallest class of structures containing $\mathcal{A}$ and closed under ultraproducts and ultraroots. This is somewhere in Chang/Keisler if I recall correctly, but that's a big book and I can't find it at the moment so instead see Fact $3.2.5$ here.)