I'm new to neurobiology so I don't know much about it. However, I have worked on artificial neural networks. Man-made AI networks all follow a handful of simple rules. I was wondering if biological neurons also have a set of predictable rules? E.g What does a neuron do if it does not get stimulated/receives no signal for a long time. Will it die ... or try to find new connections that might send signals? What does a neuron do if it is over-stimulated? What does a neuron do if it receives a signal -> does it broadcast it to all of its connections, or only 1. Does it have a preferred output path for any incoming signals. Can a neuron send out a new signal without having received a signal from anywhere else. How does a neuron know if it has done his job "well". How does it get positive feedback.
"Always" is always a dangerous term in biology.
What does a neuron do if it does not get stimulated/receives no signal for a long time. Will it die ... or try to find new connections that might send signals?
What neurons do when they aren't stimulated depends on the neuron and phase of development. Typically, thresholds for spiking will decrease, but it's also normal for cells to atrophy and die: neuronal death is a critical part of development, and loss of synaptic connections and some neurons is the most obvious feature of human brain development during adolescence, rather than birthing new neurons or growing new synapses.
What does a neuron do if it is over-stimulated?
Overstimulated neurons may die, they may also reduce the efficacy of their inputs either by increasing their thresholds or reducing synaptic weights by removing postsynaptic receptors. Calcium is an important mediator of these signals - you could read about its role in long term depression for example (this has nothing to do with 'depression' in mental health, just reduction in synaptic weight). There are also short-term effects that limit overstimulation of particular pathways, with many acts on the pre-synaptic rather than post-synaptic cell. Neurotransmitter is held in vesicles, and after release, there are fewer vesicles available for the next release event. Presynaptic terminals also often have autoreceptors that cause negative feedback and limit subsequent transmission.
What does a neuron do if it receives a signal -> does it broadcast it to all of its connections, or only 1
Of all your questions, "does it broadcast it to all of its connections" has the simplest answer: biological neurons almost always broadcast to all of its forward connections in response to any sufficient input (just like most artificial neural networks). However, the release probability at any one synapse is typically <1. Therefore, a given cell can make much higher-fidelity (and also higher-amplitude) connections with another cell by the number of contacts. If a typical cortical neuron makes only one synaptic contact with another cell, there will be a high rate of failures of transmission between the two cells. It's common for stronger connections to involve up to hundreds of individual contacts, just between two cells.
Can a neuron send out a new signal without having received a signal from anywhere else
Some neurons are spontaneously active. There are various types, but the most familiar are the ones thought of as intrinsically oscillating. An example are the sinoatrial node cells of the heart (although these are not really neurons, they act a lot like neurons; there are similar cells that are neurons that control respiration). There are also some CNS cells.
How does a neuron know if it has done his job "well". How does it get positive feedback.
This question is the most complex of all the questions you asked, entire textbooks and journals are needed to answer it, and it isn't at all possible to answer here. I'll just say that many of the learning rules used in artificial networks are not biologically plausible, they are simply computationally efficient. Biological neural networks use a lot more recurrent activity than artificial ones do, and a lot of the feedback like this occurs within that recurrent network. Biological networks are also built to do a variety of computations, and learning rules that apply in one may not apply in another. For example, pattern separation versus pattern completion require completely different learning rules.
However, the very general principle of updating synaptic weights in response to errors or outcomes does hold between artificial and biological networks.