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Note: in case you, like me, were confused by the notation, the author appears to use $A$ to refer to the entire sphere, not just the part you colored blue.

Question 1: The blue image, and all subsequent images generated at each finite stage of the process, have no holes. The blue image is the embedding of $S^2$ into $\mathbb{R}^3$ given by taking the standard one, drawing two circles on it, and "pulling" them out to create the "handles" in the image. Nothing is ripped in the process, and it can be undone by shortening the edges of the handle until the circles are again flush with the surface. (Note that this process involves "corners" as drawn, but it's easy to round them out and thus get something smooth). The process is animated here.

Question 2: You're right to be wary of the role of approaching a limit here. The Alexander Horned Sphere itself is an illustration of what can go wrong by taking a process to infinity. Note that at every finite stage of its construction process, this embedding is actually equivalent to the normal one! (Imagine smushing the handles back into the sphere to go backwards a step, and keep doing this until you get back to where you started). In particular, at each finite stage, the exterior of the sphere is simply connected. However, it's not too hard to see that the loop $L$ remains outside the sphere even in the limit. This is because at each finite stage, the amount of space in $\mathbb{R}^3$ where things are changing is smaller. You can draw a little $3$-dimensional ball $B$ around the loop $L$ such that at every stage past the first, nothing changes inside this ball. In symbols, if $A_n$ is the sphere obtained at the $n^{th}$ stage of the process, then $B \cap A_n = B \cap A_1$ for all $n \geq 1$. This means that, since $L \subset B$, $$L \cap A_n = (L \cap B) \cap A_n = L \cap (B \cap A_n) = L \cap (B \cap A_1) = (L \cap B) \cap A_1 = L \cap A_1 = \emptyset$$.

Question 3: First, we'll see why the disc $D$ matters. A loop $L$ in a space $X$ should be thought of as a continuous function $\gamma: S^1 \rightarrow X$. (Point of clarification here: we don't require $\gamma$ to be an embedding and allow self-intersections). Then, a deformation of $L$ to a point $x \in X$ is exactly a map $\Gamma: S^1 \times [0,1] \rightarrow X$ such that $\Gamma(s, 0) = \gamma(s)$ and $\Gamma(s, 1) = x$ for all $s$. You're supposed to think of it as a continuously varying family of loops $\gamma_t(s) = \Gamma(s,t)$. For each fixed $t$, changing the $s$ variable describes a loop. For $t = 1$, this is the constant loop at the point $x$. Now, since $\gamma_1(s)$ is a constant function in $s$, we can think of $\Gamma$ as actually being defined on $D$, as follows:

Using polar coordinates on $D$, we can write $d \in D$ as $(r, \theta)$ with $r \in [0,1]$ and $\theta \in S^1$. We can define $\tilde{\Gamma}: D \rightarrow X$ by $\tilde{\Gamma}(d) = \Gamma(1-r, \theta)$. Since $\Gamma(1, \theta)$ is constant in $\theta$, this is well-defined and continuous.

Thus, we may assume by way of contradiction that there is some function $F: D \rightarrow \mathbb{R}^3 \setminus A$ such that, in polar coordinates, the map $\gamma: S^1 \rightarrow X$ described by $\gamma(\theta) = F(1, \theta)$ coincides with $L$. Now, as stated in the text, there is some definite number $\epsilon > 0$ such that for every point $y \in F(D)$, the distance between $y$ and every point $a \in A$ is at least $\epsilon$. (This is because both $A$ and $F(D)$ are compact and they don't intersect by assumption). This number $\epsilon$ depends on $F$, but not on $y$ and $a$. However, we are assuming there is a certain fixed $F$ which satisfied our hypotheses. Now, for each sphere $A_n$, you can see by staring at the picture that the only way to shrink the loop $L$ to a point is to move it through the little gaps in the picture. In particular, the image of the disc given by this deformation will include some points within the little gaps. (As a hand-wavy explanation, think about the fact that $L$ has some points "under" the biggest handle and some points "over" it. In order to shrink it to a point, we have to bring the "under" parts up or the "over" parts down). Now at each finite stage, this is no problem: we can always choose a disc small enough to get through the littlest gap. However, for the full limiting construction $A$, there is no littlest gap; the handles are interwoven so tightly that any gap would be smaller than $\epsilon$ for any $\epsilon$ whatsoever. Of course, this is impossible to really visualize, so it's easier to convince yourself with a limiting argument. By drawing each $A_n$ carefully, you can arrange it such that the distance between the two sides of the little gap in $A_n$ is less than half of the distance in the previous step. Thus, the limiting size of the little gaps is less than $\frac{1}{2^n}A_1$ for all $n$, and thus it is $0$! In particular, there is some finite stage $A_n$ such that the gap in $A_n$ is less than the $\epsilon$ we fixed earlier. Since the size of the gap in the limit is smaller than the size of the gap at every finite stage, this too is less than $\epsilon$. But since $F(D)$ has to contain some points inside the little gap, it has to contain a point $y$ such that $y$ is within distance $\epsilon$ of the wall, a contradiction.

This is a strange and somewhat disturbing construction. It's a little hard to believe that this limiting object really exists, and that it is still an embedding of $S^2$. It takes a bit more argument to show that this is true; see here. (Does anyone know of a "canonical" reference for this fact?)