From $x_n\le y_n\le x_{n+2}$ for all $n\in\Bbb N$ you can infer that
$x_0\le y_0\le x_2\le y_2\le x_4\le y_4\le\ldots\tag{1}$ and
$x_1\le y_1\le x_3\le y_3\le x_5\le y_5\le\ldots\;,\tag{2}$
but there’s nothing to tie the odd and even subsequences together. In particular, it does not follow that $\langle y_n:n\in\Bbb N\rangle$ is increasing: it might be, for instance, that
$x_n=y_n=\begin{cases} n,&\text{if }n\text{ is even}\\ 1-\frac1n,&\text{if }n\text{ is odd}\;. \end{cases}$
Then the odd and even subsequences are both increasing, but $\langle y_n:n\in\Bbb N\rangle$ is bouncing up and down like crazy. This example also shows that $\langle y_n:n\in\Bbb N\rangle$ need not be bounded (though it can be), and that $\langle x_n:n\in \Bbb N\rangle$ need not be either increasing or decreasing.
I’ll just give you a hint for the remaining one: you can use $(1)$ and $(2)$ to show that if one of $\langle x_n:n\in\Bbb N\rangle$ $\langle y_n:n\in\Bbb N\rangle$ converges, then so does the other, though it’s quite possible that neither converges. Remember, a sequence $\sigma$ converges to some limit $L$ if and only if every subsequence of $\sigma$ converges to $L$.