Over what length of time would you need to collect the data, to make this an accurate survey?
You need to choose the test period to gain the most from the survey. Statistics cannot help you here as it simply says "Give me a model of your system and I can tell you the pattern that model will exhibit, and thus I can tell you for example the significance of a slightly differrent pattern". So you need to devise a model which of course must be acceptable to your audience.
Normally you would design sampling to avoid systemmatic error. So if one individual did all the load sheets for one shift then doing one shift would be less predictive of things in general than spreading the readings to cover several shifts. Unless of course you were interested in predicting the performance of that shift, in which case the reverse applies.
What would be the % error in the results?
You are simply categorizing them as right or wrong. So you would first have to check your checking. After that I would take a simplistic view. If you did 1 reading and it was 'wrong' clearly you wouldn't want to say the error was 100%. Similarly if you do 100 and 1 is wrong, you don't want to go for 1%. However if 20 were wrong you would be safe to say 'approx 20%'.
Could you then apply that same error of probability to another fleet in your airline? Would it be accurate to say for example that if we found a 20% error in loadsheets with one fleet, then the same could be applied to another fleet?
Yes - providing you think the same model applies. Most likely it doesn't so you would be best to compare your results with some others. Try a third fleet either within your company or externally
Sorry if this doesn't help but statistics tend to deal with things that vary but conform to a known distribution function. Your problem is you can't see variation (unless you did experiments on lots of fleets) so you can't easily build a model of variability.