It depends on what you are scanning. If you are attempting to use the central limit theorem then you must verify that the underlying hypotheses are indeed true if you want to prove something. Otherwise, you have to make assumptions and if too many than the proof is not very reliable.

There are other issues as well depending on what you are trying to measure. If it has some underlying geometric structure then you may have to prove a uniqueness of approximation theorem. Otherwise, your increased sampling may converge to one of many different things and replication may be a problem. For example if you take X-rays of increased frequency of some unknown object from a single point source, increasing frequency allows you to construct some approximate shape of the unknown object but what you construct could be one of infinitely many different shapes producing the same measurements. However, if you use two somewhat independent point sources AND you know the shape is convex then it is known that increasing the number of measurements for X-rays, at some uniform angular spread from each source, will lead to approximate shapes which have to converge to the true shape of the object.

The point is, your question is a bit too general. Need more specifics.

I'm not sure I understand this question correctly. Suppose our task is to make sure that some device is still in operable conditions, for example is our clock still running or its battery has to be replaced. The result of our check is either 0 (battery failed) or 1 (everything is correct). The checks may be done once a week, daily or every hour. The probability of error detection is roughly the same in each of these cases (well, it slowly increases with time as battery discharges). Increasing the frequency of checks will then result in earlier detection of failure, not its probability.

On the other hand our error detection procedure may be not perfect. If this is the case, then increasing frequency of tests will increase the probability of (correct) failure detection within prescribed time interval.

I agree. While it's often good to know more detail, this doesn't necessarily mean that you will be able to detect errors of a particular type or that the result will be more accurate. While this isn't exactly your question, I might point out that there is a significant misunderstanding between random and systematic error or uncertainty that is often overlooked. Many people like to divide the standard deviation by sqrt(n) so that it appears if you average enough data you arrive at a result with a vanishing (tending to zero) uncertainty. This is not true. Averaging a lot of inaccurate numbers won't yield an accurate result and you can't eliminate systematic error. In recent NIST documents you will see "repeatability" used more often than "uncertainty" or "error". Just because a result is repeatable doesn't mean it's accurate. History is dotted with "Pathological Science" as you will see from a Google search, including Irving Langmuir's classic text.