Question

when multiplying or dividing, the sign of the answer depends on the sign of the largest absolute value of the number.

Answer 1

Absolutely not:
Consider (-1)*5. The largest number by absolute value is 5. But the sign of the answer is - , as the result is -5.

Generally, when multiplying or dividing, the answer depends on the number of negative signs you have. If you have an even amount (multiple of 2) of negative signs (so either 0,2,4,6,8 and so on) many negative signs, the result will be POSITIVE. Should you have an uneven amount of negative signs (so either 1,3,5,7,9,11 and so on), the result will be NEGATIVE.