gn.general topology - Freeing a sphere from within a sphere

For your first question, look at complements (a "fundamental" technique in analyzing ambient isotopies/ambient homeomorphisms; e.g. the "knot group"):

Let the union of the initial two spheres be $S$, and the union of the final two spheres be $T$. An isotopy on $mathbb{R}^3$ taking $S$ to $T$ in particular ends with a homeomorphism from $mathbb{R}^3$ to itself taking $S$ to $T$, hence producing a homeomorphism from $U=mathbb{R}^3setminus S$ to $V=mathbb{R}^3setminus T$. But $U$ has only one of its three connected components contractible (the inside of the inner sphere), whereas $V$ has two of its three components contractible (the insides of the two spheres), a contradicting that they be homeomorphic.

$Big($One way to prove the other components are not contractible is that they have non-trivial second homotopy and homology groups, as exhibited by the elements represented by the spheres themselves.$Big)$

For your third (and second) question, in generality, here's an argument that there is no embedding from $M=S^{n-1}times I$ to $mathbb{R}^n$ with boundary ambient isotopic to the "side-by-side" embedding of
$S_0sqcup S_1$. By any letter $S$ I denote a copy of $S^{n-1}$. For comfort of the imagination, think $n=2$:

(1) Up to ambient isotopy, there are only three embeddings of $S_0sqcup S_1$ in $mathbb{R}^n$ : the "avocado$^+$ type" embeddings $A^+$ with $S_0$ inside $S_1$, the "avocado$^-$ type" embeddings $A^-$ with $S_1$ inside $S_0$, and the "side-by-side type" embeddings $B$ with $S_0$ and $S_1$ "next to" each other.

$Big($That they fall into these three types is just a case analysis; that any two embeddings in the same "type" are ambient isotopic is also easy: (a) first translate and dilate until the $S_1$ embeddings match up, and then being in the same case means the two $S_2's$ are embedding in the same component of the compliment of $S_1$, so (b) you can move them around in that component until they match up, too.$Big)$

(2) $A^pm$ are not ambient isotopic to $B$, by counting contractible components of their complements, as in the answer to your first question: $A^pm$ has one, $B$ has two.

(3) An embedding from $M=Stimes I$ to $R^n$ cannot have $partial M = B$ up to ambient isotopy.

$Big($ Why? Here's one proof, which I found kind of fun: Suppose $partial M = B$, i.e. is in the "side-by-side" arrangement. Since $M$ is connected, $int(M)=Msetminus partial M$ must lie outside both spheres of $partial M$, since if it lies inside one, it can't touch the other. Let $x$ be the center of of the $S_1$ component of $partial M$. Now $M$ is compact, so it has a point $p$ which is of maximal distance from $x$. Such a $p$ must be a boundary point of
$M$, since at an interior point, we could move slightly in a direction away from $x$ and stay inside $M$. But $p$ can't be on $S_1$ or $S_2$ either, since their outer sides are "padded" by the open set $int(M)$... this a Euclidean geometry argument which I'll stop rigorizing here. So $p$ is nowhere, a contradiction.$Big)$

(4) By process of elimination, $partial M$ must be ambient isotopic to $A^pm$, which by (2) is not ambient isotopic to the "side-by-side" embedding $B$, which is what you wanted.