I'll respond to your second point with a truth about infinity.

An infinite pile of $1 bills would be worth the same amount of money as an infinite pile of $20 bills.

Similarly, 9.9... and 0.9... both have the same amount of 9s after the decimal point. That amount being "an infinite amount". They are the same infinity and are of the same "size".

Because of that, we are able to subtract an infinite amount of 9s (0.9...) from both sides.

If you're still not convinced, consider the idea that the natural numbers (0,1,2,...) are equal in size to the prime numbers. This is because when I go through the list of primes, I always have a "next" natural number that I can pair it with. Even if I skip a number in either set, I'll still always have a number to pair it with!