Evolution is not a mechanism that maximizes it is a set of interrelated elements that operate at entirely different levels (molecular, cellular, individual and species) to pass on those traits from one generation to another that result in (possible) change of future individuals within a population, which may affect the survival of those individuals increasing or decreasing the chances of passing those traits on to their (hopefully viable!) offspring. It does nothing to ensure survival of any particular individual, at best it may help a trait that has a function in survival to be passed on.
Mutations don't 'succeed' they get passed on or they don't, usually without an immediate effect on the individual, especially not in the individual where the mutation first takes place. But over a longer period mutations may result in a statistical advantage against some environmental factor (including predators) resulting in an increased chance of that particular mutation becoming more widespread. The result is possibly that that mutation ends up being carried by all individuals of the species but that takes a relatively long time and rarely results in 100% coverage in the first generations if at all.
ANN's do not necessarily 'teach themselves', there are supervised ways of using ANNs and there are unsupervised ways of using them.
> They get an answer wrong, numbers shift and if those numbers help the next instance they stay or shift as needed.
No, they optimize for the statistical likelihood of getting all of the answers right across the whole training set, and then we check if it worked by verification with held back data. So it is very well possible that a change in the weights negatively affects some outputs but positively affects others, if the change is a 'net positive' we keep it. To make matters more interesting every now and then we throw away all but a fraction of all the weights.
In the case of supervised learning we (humans) label the data and are the 'teachers' and in the case of unsupervised learning the ANN does its own labeling of clusters of related data (or, more accurately, clusters of data that seem to be related).
There are also intermediary versions where humans do some of the labeling to set up the initial clusters and then the mechanical labeling takes over to rapidly increase the size of the training set taking into account the possibility of getting some of the training data wrong but still coming out ahead across the board.
So yes, in ANNs there is a global maximization around a goal, and we set it up that way. Evolution doesn't have a particular goal, it is a result of a number of interrelated factors not an input and not something that was put together with a goal in mind.