1. This is an example of an expression that has altered its spelling over time. "Not for donkey's ears" means not for a very long time and refers to the length of a donkey's ears. However, they gradually stretched from the "donkey" and turned "ears" into "years". The meaning of the expression is the same, if it does make rather less sense than the original.
2. Because donkeys tend to plod along slowly, they take a long time to complete a task or journey. Therefore, a donkey year takes a long time to elapse.
I haven't heard that one for donkey's ears.