Isn't it also possible for two numbers to have unequal expansions but still be equal? Take .3... * 3 vs 1. The first expands to .9... while the second expands to 1.0... but they are also equal.
Right, but note that we can use numbers from countable sets and find different symbolic representations for them (3.0 vs 3). That problem is not inherent from the uncountability of the real numbers.
Yup, it is possible. That was just a hand wavy example to show where the issue is. Symbolic real numbers need not be represented as small programs/objects generating a sequence of digits. But all of those implementations/representations will fail to compare some numbers.
Edit: for instance, you could imagine a representation (think object of a virtual class in c++) that gives you a fraction and a 1/2^n bound for error, where you can query the number for whatever n you see fit. Still, in some cases it's undecidable to determine which number is greater.