For the first question, given a sharp monoid $P$ (i.e., the group of units of $P$ is the
zero group), there is a log structure on any scheme $X$ given by $M_X={\mathcal O}_X^*\oplus P$ with the structure map given by $(f,p)\mapsto f$ if $p=0$ and $(f,p)\mapsto 0$ if $p\not =0$. This comes from the prelog structure $P\rightarrow {\mathcal O}_X$ given by
$p\mapsto 1$ if $p=0$, $p\mapsto 0$ if $p\not=0$. I wouldn't call this a stupid log structure: it in fact can be quite useful. For example, the standard log point is defined as the log structure on the scheme $Spec(k)$ with $P$ the natural numbers as above.

For the second question, I'm not quite sure what direct sum decomposition you are expecting, but I can't think of a natural direct sum decomposition here. If you are thinking that
$p_*M_Y$ will decompose as a direct sum of the sheaf ${\mathcal O}_X^*$ and a sheaf supported
on the branches $x$ and $y$, this is not the case. To take a simpler example, taking
the divisorial log structure associated to $0 \in \mathbb{A}^1$, the log structure does
not decompose as $O_{A^1}\oplus {\mathbb N}_0$.
Indeed, there is no
section of $M_{\mathbb{A}^1}$ with support at $0$. In the example you describe, there will
be an exact sequence
$$
1 \rightarrow {\mathcal O}_Y^*\rightarrow M_Y \rightarrow {\mathbb{N}}_x \oplus {\mathbb{N}}_y\rightarrow 0
$$
where the subscripts $x$ and $y$ denote which branch the constant sheaf with coefficients
${\mathbb N}$ is sitting on. I believe when pushing down, one obtains an exact sequence
$$
1\rightarrow {\mathcal O}_X^*\rightarrow p_*M_Y\rightarrow {\mathcal F} \rightarrow 0
$$
where ${\mathcal F}$ is the sheaf supported on $D$, the vanishing locus of $t$, which
is obtained by taking the constant sheaf ${\mathbb N}$ on $D-\{0\}$ and extending by zero
across $0$. This is because there are no functions on the inverse image of a open neighbourhood of $0$ which do not vanish off the two branches, but do vanish one or both of the branches.