Yes, a neuron "firing" means it depolarizes. That is, sodium is allowed to enter the cell (ultimately resulting in influx of calcium to release neurotransmitter). read more
The transporters involved in removing neurotransmitter are secondary active transporters. This means that they do not directly consume ATP, but utilize the electrochemical gradient of other ions (e.g. sodium), which then has to be restored by ATP consuming processes. read more
"Learning" in a neuron is, from a molecular point of view certain changes in molecular compositions and localization that alter how the neuron reacts, based on what has previously happened to it. This can generally happen two places, pre-synaptically (on the neuron that"fires" or releases neurotransmitter chemicals) or the post-synaptic (the neuron that detects the released neurotransmitter) neuron. read more
The neuron was left less than fully repolarized, making it more prone to fire and depolarize again. This led to what the authors termed a “deadly loop” as the ion pumps kept trying to repolarize the membrane, sucking up more ATP, leading to even lower ATP stores and worsening the situation. read more