The second part is the paradox of Thomson's lamp, which can be formalized as thinking about Grandi's series $\sum_{n=0}^{\infty}{(-1)^n}$. This is a decent summary; one can argue either "the sum is 1/2" or "there is no sum" based on the formalization used.
For some philosophers, this is a good illustration that the operation above does not occur in real life (which is not necessarily irrelevant, since it affects which models of space and time are valid). The research generally falls under the title of "supertasks".
To address formalizing the first part of the problem, I prefer the analogous
$\sum_{n=0}^{\infty}{10^n} \sum_{n=0}^{\infty}{10} - \sum_{n=0}^{\infty}{1^n}$sum_{n=0}^{\infty}{1}$
so that instead of considering the series
1 - 1 + 1 - 1 + 1 - 1 ...
we are thinking about
10 - 1 + 10 - 1 + 10 - 1 ...
although there are certainly other ways. One can try to obtain the continuation of the function $f(1-\frac{1}{2^n})=9n$ from [0,1] using the Alexandroff compactification of the real numbers. This gives the answer of "infinite", although I'm not keen on this formalization because it removes the iterative sense of the original problem.
I find such paradoxes to be most useful (to mathematicians) pedagogically, because it allows students to apply their mathematical intutition and give them some investment before formalizing their handling of infinity.
