Let $M^n$ be a smooth closed, oriented $n$-manifold. Let $S_0,S_1\subset M^n$ two connected, compact and (positively) oriented submanifolds of $M$ of codimension $k$ diffeomorphic to $S$.

1. Suppose $k=1$, $\partial S_i = \emptyset$, under which assumptions $S_0$ is isotopic to $S_1$? Is there a set of complete invariants, e.g. refining the homotopy class $[S_i] \in [S, M]$ ?

I am aware this is a general question. However, this should be easier than the usual knot-theoretical setting (i.e. $k=2$). For example, if $\dim M = 3$, and $\pi_1 S_0\hookrightarrow \pi_1 M$ then it's enough for $S_1$ to be homotopic to $S_1$ for being isotopic see Ian Agol's answer. References are welcome too.

2. Suppose $k = 0$, $\partial S_i \neq \emptyset$. Under which assumptions $S_0$ is isotopic to $S_1$? What is an example when they are not isotopic? Is this case easier or more difficult than the $k=1$ case?

Here again, my hopes for a classification stems from Palais' theorem which asserts that if $S_i\simeq \mathbb{D}^n$ then $S_1$ is always isotopic to $S_0$. So in some degree of generality we have a nice answer.

I'm particularly interested in the following weaker case:

3. Suppose $k=0$, and $S_i$ have diffeomorphic complement, i.e. $M\setminus \mathrm{int}(S_0)\simeq M\setminus \mathrm{int}(S_1)$, can we conclude that $S_1$ is isotopic to $S_0$?

I expect this extra hypothesis to be strong similarly to the case of knots in $\mathbb{S}^3$.