  Let $\mathbb{F}_{2^m}$ be a finite field of $2^m$ elements, and
$R=\mathbb{F}_{2^m}[u]/\langle
u^k\rangle=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$
($u^k=0$) where $k$ is an integer satisfying $k\geq 2$. For any odd positive
integer $n$, an explicit representation for every self-dual cyclic code over
$R$ of length $2n$ and a mass formula to count the number of these codes are
given first. Then a generator matrix is provided for the self-dual and
$2$-quasi-cyclic code of length $4n$ over $\mathbb{F}_{2^m}$ derived by every
self-dual cyclic code of length $2n$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$
and a Gray map from $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ onto
$\mathbb{F}_{2^m}^2$. Finally, the hull of each cyclic code with length $2n$
over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ is determined and all distinct
self-orthogonal cyclic codes of length $2n$ over
$\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ are listed.
