How do I obtain recordings of the P300 wave of the event-related potential in the EEG? I am trying to understand how to analyze the ERP (Event-Related Potentials) from EEG recordings and in particular  the P300 wave.

I have come up with a few questions which I hope you might be able to help with:


I assume that the number of repetitions of the stimulus should give better results, but I am not sure why. On the one hand, it should help, because averaged results are usually better in electrophysiology. On the other hand, maybe there is some unwanted adaptation effect in the subject? So, are repetitions wanted in this kind of experiment?
There are time points in which I didn't expect P300 but the brain activity was not 0. I understand that there is always a background activity in the brain, but maybe there is more to it? And if so, are there certain frequencies that are expected in the EEG?
When I analyze the data using an average response, should I assume anything on the P300 response?
What is the recommended sample Rate I should use? I guess there should be an advantage for high sample rates to increase resolution, but I am not sure about that. Am I right, or maybe a lower sample rate has advantages that I am not aware of?
The P300 wave is a positive deflection in the human event-related potential (ERP). A common experiment in which it is analyzed is the "oddball" paradigm, where a subject detects an occasional target stimulus in a regular train of standard stimuli. The P300 wave only occurs if the subject is actively engaged in the task of detecting the targets and its amplitude varies with the improbability of the targets. Its latency varies with the difficulty of discriminating the target stimulus from the standard stimuli. A typical peak latency when a young adult subject makes a simple discrimination is 300 ms. The P300 wave may represent the transfer of information to consciousness, a process that involves many different regions of the brain (Picton, 1992).

Your questions:


ad. 1: Averaging reduces the random background EEG activity (noise), as well as artifacts due to movement or eye blinks. Random events will average out, while the ERP itself, being highly synchronized with the stimulus, will persist in the signal. In short: averaging increases the signal-to-noise ratio (SNR). Is this wanted? - That depends on the SNR you wish to obtain. The more averages, the better the SNR. [M]aybe there is an unwanted adaptation effect of the subject - if you are afraid of adaptation - do repeated measures and analyze single ERPs, or small chunks of averages across the recording - it will give you an idea whether there is adaptation, and if yes, how much. I wouldn't worry about it too much. The long time scale of P300 recordings (second-range) allows the nervous system plenty of time to recover.
ad. 2. There is always background EEG, except in (brain) dead subjects - and these subjects won't be too cooperative. [A]re there certain frequencies that are expected in the EEG - ERPs are not analyzed in the frequency domain, but in the time domain. If you are interested in the frequency content of the EEG; it is much dependent on the state of the subject. An awake, vigilant test subject with their eyes open will show beta activity (6 - 31 Hz band).
ad. 3. Averaging doesn't change the timing characteristics of the ERP, barred the recording is properly time-locked with the stimulus.
ad. 4. Higher sampling rates are generally better. If your system allows it, crank it up. You can always downsample later. Downsampling may have the advantage of smoothing high-frequency noise and enhancing the ease of handling of the data. ERPs are slow responses, so you don't need 100 kHz signals. For EEG generally 500 Hz is used, because according to the Nyquist criterion you need at least twice the numbers of samples than your highest frequency of interest. The gamma band is 32+ Hz, so 500 Hz is pretty safe (De Haan, 2013). For ERPs, 200 Hz may be sufficient. Wikipedia suggests higher rates for EEG. 


References
 - De Haan, Infant EEG and Event-Related Potentials; Psychology Press (2013)
 - Picton, J Clin Neurophysiol (1992); 9(4): 456-79