Possible mistake in Bloch Book: $\Bbb S^2\#\Bbb S^2\#\dots\cong\Bbb S^2$!

The following is in the Bloch's Book: A First Course in Geometric Topology and Differential Geometry:

and I couldn't find this in errata list. Is that correct? I found two posts here and here that indicate this is an obvious mistake.

Q2: What is the exact definition/realization of the infinite connected sum? I think one candidate could be like this: $$M_1=M, M_2=M\# M,\quad M_3=M\# M\# M,\quad\dots\quad M_n=M\# M\#\dots\# M$$ and tend $n\to \infty$ that seems that this is not the case. why?

I agree with the first post you link that connect summing with an infinite amount of spheres would be the same as removing a disk, so as written the argument is not correct.

You can fix the argument though. We would have that $\mathbb{R}^2 \cong A - D^2$, so you just need to show that if $A$ is not a sphere $A -D^2$ can't be homeomorphic to $\mathbb{R}^2$. This follows from an application of what is called the Alexander trick (and it applies to all dimensions). The Alexander trick says you can extend a self homeomorphism of the boundary of a disk to the entire disk.

However, it does not work smoothly. But this is because the result you want to prove actually is not true smoothly. There exist exotic spheres which have inverses under connect sum.