Chromatic t-structures?

The second question turns out to have a surprisingly easy negative answer. This is depressing on two counts: both that the answer is negative and that it's so easy.

Suppose that $Sp_{K(n)}$ has a $t$-structure such that $\mathbb S_{K(n)} \in Sp_{K(n),\geq 0}$. Let $F(n)$ be any finite type-$n$ $p$-local spectrum. Then for some $k \geq 0$ we have that $\Sigma^k F(n)$ is in the closure of $\mathbb S_{(p)}$ under finite colimits in the category $Sp_{(p)}$ of $p$-local spectra. Therefore, $\Sigma^k F(n)_{K(n)} \in Sp_{K(n),\geq 0}$. But we also have $\Sigma^k F(n)_{K(n)} \simeq T(n)_{K(n)}$, which is a periodic spectrum. Thus $\Sigma^l F(n)_{K(n)} \in Sp_{K(n),\geq 0}$ for all $l \in \mathbb Z$. Since $F(n)$ was an arbitrary finite type-$n$ spectrum, we see that all $K(n)$-localizations of finite type-$n$ spectra are in $Sp_{K(n),\geq 0}$.

Now I'm pretty sure that every object of $Sp_{K(n)}$ is a colimit of $K(n)$-localizations of finite type-$n$ spectra. It follows that every object is in $Sp_{K(n),\geq 0}$ and the $t$-structure is trivial. But I can't find a reference for this fact at the moment, so here's an alternate argument. It's at least the case that $\mathbb \Sigma^l \mathbb S_{K(n)}$ is a (sequential) colimit of $K(n)$-localizations of finite type-$n$ spectra for all $l \in \mathbb Z$, and so $\Sigma^l \mathbb S_{K(n)} \in Sp_{K(n),\geq 0}$ for all $l \in \mathbb Z$. If the $t$-structure is monoidal, it follows that $Sp_{K(n),\geq 0}$ is closed under desuspension, i.e. $Sp_{K(n),\geq 0} \subseteq Sp_{K(n)}$ is a a stable subcategory. This kind of $t$-structure is not very interesting, and anyway I believe that since $Sp_{K(n)}$ doesn't admit any nontrivial Bousfield localizations, it doesn't admit any such $t$-structures which are nontrivial either.

To expand on Tim's answer, the arguments generalize to show that $Sp_{K(n)}$ admits no non-trivial t-structures in general.

The crucial ingredient is that $Sp_{K(n)}$ has no non-trivial localising or colocalising subcategories, see 7.5 in Hovey-Strickland. Thus, to finish the argument it is enough to show that any subcategory $C \subseteq Sp_{K(n)}$ which is closed under limits is in fact colocalising, ie. it's also closed under suspension.

If $X$ is any $K(n)$-local spectrum, then it is well-known (see 7.10 in the aforementioned book) that it can be written as a limit $X \simeq lim \ X \wedge F_{i}$ of its smash products with type n generalized Moore spectra. If $X \in C$, then some desuspension of $X \wedge F_{i}$ is contained in $C$ as well, but as Tim observed these spectra are periodic and so $\Sigma^{n} X \wedge F_{i} \in C$ for all $n \in \mathbb{Z}$. It follows that $\Sigma^{n} X \in C$, ending the argument.