measurement errors: add up or cancel out?
You always have two kinds or errors: Random error and systematic error

Systematic errors are there always with the same amplitude, they don't change (over long periods of time and extanded temperature range they do have drift as well, this will be defined e.g. in ppm /°C or ppm /y). This means these errors won't cancel out with averaging, because all measurements will have the same systematic error. But if you can determine the systematic error (e.g. by some calibration routine) you can compensate for this error by subtracting a calibration value from your measurement result.

Random errors behaves like you descripe, every measurement varies by some degree due to noise from all kind of sources. You can not correct this error from a single measurement, because there is no way you could possibly know how this exact measurement was altered. BUT you can average this random error down. If you take an infinite amount of samples your average random error will approach 0 (you can still have a systematic error, meaning the mean of your measurements is not representing the real mean value).