  Let $R=\mathbb{Z}_{4}[v]/\langle
v^2+2v\rangle=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^2=2v$) and $n$ be an odd
positive integer. Then $R$ is a local non-principal ideal ring of $16$ elements
and there is a $\mathbb{Z}_{4}$-linear Gray map from $R$ onto
$\mathbb{Z}_{4}^2$ which preserves Lee distance and orthogonality. First, a
canonical form decomposition and the structure for any negacyclic code over $R$
of length $2n$ are presented. From this decomposition, a complete
classification of all these codes is obtained. Then the cardinality and the
dual code for each of these codes are given, and self-dual negacyclic codes
over $R$ of length $2n$ are presented. Moreover, all $23\cdot(4^p+5\cdot
2^p+9)^{\frac{2^{p}-2}{p}}$ negacyclic codes over $R$ of length $2M_p$ and all
$3\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p-1}-1}{p}}$ self-dual codes among them are
presented precisely, where $M_p=2^p-1$ is a Mersenne prime. Finally, $36$ new
and good self-dual $2$-quasi-twisted linear codes over $\mathbb{Z}_4$ with
basic parameters $(28,2^{28}, d_L=8,d_E=12)$ and of type $2^{14}4^7$ and basic
parameters $(28,2^{28}, d_L=6,d_E=12)$ and of type $2^{16}4^6$ which are Gray
images of self-dual negacyclic codes over $R$ of length $14$ are listed.
