@a tiger has given a nice answer, but to me it seems too summarized. So I just want to elaborate it by giving some more details. Also, let me tell you at first that it is a very broad topic, so covering up everything in anything less than a book is pretty much impossible. I'll only add points enough for an introductory answer. I will answer your questions one by one.
Are some "actions" or "transmissions" more powerful than others, contributing more dramatically to action potentials of neurons receiving neurotransmitters when it fires?
Yes, that is why they are called graded potentials. Graded potentials are actually different from normal action potentials in that action potentials follow the all or none law whereas the strength of a graded potential can vary depending on how many ion channels it stimulates. You might better understand this by the equations they follow.
Normal action potentials follow Nernst Equation i.e.
$E_{eq,K^+} = \underline{RT}\times \ln \underline{[K^+]_{out}}$
$\hspace{20mm}nF\hspace{12.5mm}[K^+]_{in}$
On the other hand, graded potentials follow the Goldman Equation i.e.
$E_{m} = \underline{RT}\times \ln \underline{P_K[K^+]_{out} + P_{Na}[Na^+]_{out} + P_{Cl}[Cl^-]_{in}}$
$\hspace{14mm}F\hspace{14.5mm}{P_K[K^+]_{in} + P_{Na}[Na^+]_{in} + P_{Cl}[Cl^-]_{out}}$
which is significantly different from Nernst equation. The biggest difference is that as per Goldman equation, the net potential depends on the concentration of more than one ion, meaning that it can attain a range of values, unlike that from Nernst equation. This directly answers your question that Yes, some graded potentials can be different from others in strength and time, thus giving different contributions to action potential on the following neuron. Yes, time too. It is so because different ion channels remain in different states (open, close, inactivated) for different times and take different times for transitions. You might understand this by this diagram:

How does the bias occur/form over time? Is a bias based on short term or long term series of interactions?
Generally, bias is formed by continuous strengthening of a synapse based on its activity over recent time, a process known as LTP or Long Term Potentiation. LTP can lead to a long-lasting strengthening of a synapse and is hence considered to play a major role in formation of memories in brain. When a synapse is repeatedly stimulated over time, the following neuron creates more dendritic receptors so that more effective potential can be generated. Over time, the first neuron starts releasing more and more neurotransmitters whenever it is excited. These two factors, when combined, lead to stronger synaptic connection i.e. formation of bias. You can see the following images for better understanding:




In fact, a bias results from both short term and long term potentiation. The former i.e. STP is a transient increase in synaptic strength of two neurons usually due to increase in synaptic vesicle release probability. It can also be caused by diseases such as tetanus. So, if you look at the big picture, then yes, a bias is the result of both short term and long term interactions between two or more neurons.
How many past interactions are significant in the current state of the bias at any given time?
Bias formation is a long process which requires many steps, including gene expression and protein synthesis. That is why, LTP has been divided into two (and sometimes three) different phases: E-LTP (early phase or LTP1) and L-LTP (late phase, further divided into LTP2 and LTP3). A schematic diagram of LTP is below:

Thus, in short, we don't know exactly how long it takes for LTP formation, and there probably is no exact time required for its formation.
Is there bias at all?
I should have answered this question at first, but since I do so at last, I will add some more points. Biasing, or synaptic plasticity, is the ability of synapses to strengthen (by LTP) or weaken (by LTD) over time due to change in their activity patterns. Its mechanism has been theorized to be dependent on:
- change in probability of glutamate release.
- insertion or removal of post-synaptic AMPA receptors.
- phosphorylation and dephosphorylation inducing a change in AMPA receptor conductance.
The third one is a more complex part of the process, while the first two processes have been examined by a mathematical formula for calcium-based model of plasticity as:
$\underline{dW_i(t)} = \underline{\{\Omega([Ca^{2+}]_i) - W_i\}}$
$\hspace{3mm}dt\hspace{22mm}\tau([Ca^{2+}]_i)$
where:
- $W_i$ is the synaptic weight of the $i^{th}$ input axon
- $\tau$ is a time constant, dependent on the insertion and removal rates of neurotransmitter receptors, which is dependent on $[Ca^{2+}]$
- $\Omega$ (the concentration of calcium) is also a function of the concentration of calcium ($\Omega = \beta A^{fp}_m$) that depends linearly on the number of receptors on the membrane of the neuron at some fixed point.
So in short, Yes, biasing does exist in neurons and plays an important part in many important phenomena such as memories.
P.S. I also want to add about relation of synaptic plasticity with epilepsy as you talked about in your question. Though we don't know everything about epilepsy, we know that synaptic plasticity indeed does play a role in epileptic seizures. During epilepsy, seizure threshold i.e. amount of stimulus required for action potential, is lowered. Also, because of factors such as long term potentiation, even a small potential gets enhanced into a larger potential. All these factors, along with some other unknown ones, when combined, lead to a seizure or attack.
Also, epilepsy is also caused due to decreased resistance of neurons towards a stimulus and a decrease in the refractory period of ion channels (what you call recharge time). Due to decrease in this recharge time, neurons are able to generate kind of unnecessary action potentials which ultimately leads to a seizure. As said, refractory period is the time immediately after depolarization of ion channels during which it is unable to transmit another signal. A simple graph shows refractory period like this:
