Additionally, the statement that you "must" start multiplication, addition, or subtraction from the right is simply false, (in fact many of the worlds fastest "human calculators" add and subtract from left to right). The only reason to go from right to left in our standard computation method is to "carry" to the next largest place value. In the case of something like .123 by 3. Try it from right to left and left to right. Because you don't have to "carry" it's just as easy to compute it in either order. For .33333... times 3 for example, since you have "3s forever" in the repeating decimal, you are guaranteed to multiply 3 in some decimal place value times the whole number 3, thus no carrying. This makes it perfectly okay to multiply from left to right, even in our standard "vertical" method of computation.
The assumption that there is a unique way to represent a number in decimal form is the main flaw in the argument for .9 repeating to be worth a different value that 1. This is simply not true. The decimal system is based on fraction fractions (that's why the place values are named as tenths, hundredths, and so on). There are equivalent fractions that use different numbers to represent the same quantity, and, with repeating decimals, equivalent decimals using different numbers to represent the exact same values. On the topic of fractions .9 repeating can be expressed as an infinite sum of 9*(sigma) (1/10)^n where n ranges from 1 to infinity. In calculus this sum is easily proven to equal exactly 1. Q.E.D. as it were.