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][1].
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][2] in $\mathbb{S}^3$.


  [1]: https://mathoverflow.net/questions/103395/isotopy-in-3-manifolds
  [2]: https://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0965210-7/S0894-0347-1989-0965210-7.pdf