  Let $f$ be a newform of weight $k$ and level $N$ with trivial nebentypus. Let
$\mathfrak{p}\nmid 2N$ be a maximal prime ideal of the coefficient ring of $f$
such that the self-dual twist of the mod-$\mathfrak{p}$ Galois representation
of $f$ is reducible with constituents $\phi,\psi$. Denote a decomposition group
over the rational prime $p$ below $\mathfrak{p}$ by $G_p$. We remove the
condition $\phi|_{G_p} \neq \mathbf{1}, \omega$ from [CGLS22], and generalize
their results to newforms of arbitrary weights. As a consequence, we prove some
Iwasawa main conjectures and get the $p$-part of the strong BSD conjecture for
elliptic curves of analytic rank $0$ or $1$ over $\mathbf{Q}$ in this setting.
In particular, non-trivial $p$-torsion is allowed in the Mordell--Weil group.
Using Hida families, we prove a Iwasawa main conjecture for newforms of weight
$2$ of multiplicative reduction at Eisenstein primes. In the above situations,
we also get $p$-converse theorems to the theorems of Gross--Zagier--Kolyvagin.
The $p$-converse theorems have applications to Goldfeld's conjecture in certain
quadratic twist families of elliptic curves having a $3$-isogeny.
