I will try to answer Q1.
One simple reason why $A_\infty$-algebras appear is that usual DG-algebras are not homotopy stable, in the sense that if $A$ is a DG-algebra and $V$ is a complex which is a deformation retract of $A$ (in the usual topological sense, using the notion of homotopy for complexes), then $V$ does not inherit a DG-algebra structure, but well, an $A_\infty$-algebra structure. In fact, it suffices, that there is a map of complex $V\to A$ that is right homotopy invertible, see this paper. One usually likes to do this when $V$ is quasi-isomorphic to $A$ via some map $V\to A$ (for example, $V=H(A)$), and one can produce a retraction for example when the ground ring is a field.
Now one problem which $A_\infty$-algebras fix is that of localising the category of DG-algebras at quasi-isomorphisms. If $A$ is an $A_\infty$-algebra, one can consider its bar construction $BA$, which is a honest DG-coalgebra, and take $\mathsf{DASH}$ the category with objects the $A_\infty$-algebras and hom-sets the maps DG-coalgebra maps between bar constructions. Every DG-algebra is an $A_\infty$-algebra, and one can think of the category $\mathsf{DA}$ of DG-algebras sitting inside $\mathsf{DASH}$ as a (non-full, and that's the jist of it all) subcategory via the obvious inclusion. There is a usual notion of homotopy between maps of DG-coalgebras which yields the homotopy category $\mathsf{DASH}/\sim$, and it turns out the map $\mathsf{DA} \to \mathsf{DASH}$ gives an equivalence of categories $\mathsf{DA}[\text{Qis}^{-1}] \to \mathsf{DASH}/\sim$. This is explained, and I suppose first proven in this paper by Munkholm. Thus, $A_\infty$-algebras "fix" the problem of quasi-isomorphism of complexes by replacing perhaps a big complex by a small one, usually its homology, at the cost of giving you an ugly differential.
To get back to Q1, note that in the category $\mathsf{DC}$ of conilpotent DG-coalgebras, we also have a notion of quasi-isomorphism. Thus, say that a map $f:A\to A'$ of $A_\infty$-algebras is a weak equivalence if its image in $\mathsf{DASH}$, i.e. $Bf : BA\to BA'$, is a quasi-isomorphism. It turns out that, because $BA$ and $BA'$ are quasi-free (i.e. free as coalgebras, with some funny differential), weak equivalences are homotopy equivalences (there is a model category on $\mathsf{DC}$ where all objects are cofibrant and the fibrant objects are the quasi-free coalgebras, as proven here), and in fact one can show that $Bf$ is a quasi-isomorphism if and only it is an homotopy equivalence, if and only if $f_1$ is a quasi-isomorphism. This means that, so far as quasi-isomorphism (equivalently, homotopy) of $A_\infty$-algebras goes, the only relevant map is the one of length $1$. You can find a proof of this here. As mentioned in the comments, the implication "$f_1$ a quasi-isomorphism then $Bf$ a quasi-isomorphism" is a simple spectral sequence argument, but the reverse implication is a bit more involved, and is proven in loc. cit. with comparison theorems (Zeeman/Moore) of spectral sequences and constructions (Cartan).