(What follows is largely the result of digging around online, based on knowing a few more magic words than the OP.)

**Answer to the first question (I think).**

Let $V$ be a separable Banach space. The standard proof that $V$ is isometrically isomorphic to a linear quotient of $\ell_1$ works by choosing a countable set $X$ that is a dense subset of the unit sphere of $V$, defining $f:\ell_1(X) \to V$ in the obvious way, and then verifying two things:

(i) $f$ is surjective;

(ii) the natural map $f_1: \ell_1(X)/\ker(f) \to V$ is an isometry.

Suppose $V$ is actually a unital Banach algebra. Let $S$ be the multiplicative subsemigroup of $V$ generated by $X$; of course $S$ is countable. Although $S$ might not lie in the unit sphere of $V$, it does lie in the unit ball, and so we do get a norm-decreasing homomorphism $h: \ell_1(S) \to V$.

We may factorize $f$ as $h\circ \iota$ where $\iota: \ell_1(X)\to \ell_1(S)$ is the natural isometric embedding induced from $X\subset S$. Since $f$ is surjective, so is $h$. Moreover, since $f_1$ is an isometry: given $b$ in the unit ball of $V$ and $K>1$ we can find $\xi\in \ell_1(X)$ with $f(\xi)=b$ and $\Vert\xi\Vert_1 < K$. So $b=h(\iota(\xi))$ where $\Vert\iota(\xi)\Vert_1 < K$. This shows that $h$ is actually a quotient map of Banach spaces, as required.

**Comment on the intermediate question.** I am not sure right now what one can say about the Banach algebra quotients of these algebras in general.

**Partial answer to the final question.** Via the Gelfand transform we can regard $\ell_1({\bf Z}_+)$ as an algebra of continuous functions on the closed unit disc, and hence for any closed subset $E$ of the unit disc, there is a norm-decreasing homomorphism $r_E: \ell_1({\bf Z}_+)\to C(E)$ given by restriction of these functions. It is known that for some infinite closed $E\subset {\bf T}$ this restriction homomorphism $r_E$ can be surjective: according to remarks in Volume 2 of Hewitt&Ross, these were originally called Carleson sets, but Wik showed that this coincides with the more widely known notion of a Helson subset of ${\bf T}$, which corresponds to $r_E: \ell_1({\bf Z}) \to C(E)$.

MR0125404 (23 #A2707) I. Wik, *On linear dependence in closed sets.* Ark. Mat. 4 1961 209–218.

The condition that $r_E:\ell_1({\bf Z})\to C(E)$ be a quotient map of Banach spaces corresponds to the classical notion of $E$ being a subset of ${\bf T}$ with Helson constant one. Such sets are known to exist, but I confess I don't know anything about the proofs/constructions. Wik's result shows that Helson sets are Carleson sets, but it is not clear to me from his proof if "1-Helson sets" are automatically "1-Carleson".

So I can't answer to your question in the $\ell_1({\bf Z}_+)$ case, but the answer is positive in the $\ell_1({\bf Z})$-case.