So my take is that the reals as a set doesn't have an operation for subtracting two actual infinities, and that's why you get a problem. (So essentially the subtraction of two infinite sums is not a well defined operation.) Some of your answers were along these lines but not quite all the way there.
The problem also occurs on the other side (i.e 0.9999.... = 1) but is not considered a problem because it does not give an absurd result. Or does it? I would argue that it is an absurd result: Logically 0.999... is not the same number as 1, they differ by an infinitely small non-zero amount: an infinitesimal amount. This proof "proves" that they differ by exactly zero which contradicts the premise that they are two different numbers. So if this argument is true then 0.9999... and 1 are the same number. Taken to exhaustion this concept implies that every number is the same number. I.e. If I subtract an infinitesimal amount from 0.99999... then that's still equal to 0.99999... and that's still equal to 1. I can subtract an infinite number of infinitesimal amounts and the answer will still be 1. Therefore every number is 1, that's an absurd conclusion if you ask me.
Another way to think about this is that actual infinity (i.e. infinity as a complete finished number not a concept) shouldn't really be in the number system at all. But if you are going to have it in the number system then you also need to balance it with infinitesimals.
Here's a short proof I've written which demonstrates how the reals can be supplemented with a single infinitesimal to fix the problem. Note that I use a mathematical form called dual numbers which is commonly used in physics proofs (such as perturbation theory) however it is not part of mainstream ZFC (the math you will learn in undergraduate math courses at university.)
If nothing else this little problem serves as a reminder that mathematics has never been anything more than a set of logical arguments and therefore your premises will always determine your conclusions.
Note that in this proof there is a logical jump at the first green squiggle which is a bit tricky and might not be strictly consistent with our formal definition at the top. Essentially the argument is this: If you put a 9 in every column of a base 10 number, for an infinite number of columns, then this is the largest number you can possibly express using base 10 arithmetic. (Even any sort of power etc will result in a different number which doesn't have 9's in every column and will therefore be smaller.) If you invert this number then you retrieve the smallest number you can possibly express that isn't zero. Logically that number will also be our infinitesimal.
