No, the British Museum did nothing with the Arizona results that was not done with the results from the other laboratories. Each of the twelve samples (four at Tucson, five at Zurich and three at Oxford) was tested several times as part of the process, in order to derive a measurement for error as much as anything else, and standard practice is to produce a single average result for each sample. Even today, publications of radiocarbon results do not list every single measurement. It is usually not necessary, although in this case, in view of the subsequent furore, perhaps it would be a good idea. The fact that the the other two labs did all their tests of each sample in one go meant that their results had already been averaged, while because the Tucson lab took their series of measurements of each sample in two batches, they had to be averaged into one measurement for each sample to conform with the others.
As for the actual calendar date, the dates measured by the AMS machines are directly related to the proportion of C14 to C12 in the sample, and do not take into account fluctuations in the atmospheric concentration of C14. To turn them into calendar dates they have to be calibrated against a curve derived from dendrochronological studies. This can be done by anybody, as not only the calibration curve, but even an app to help you work out the error-bars, is available online.
[edited to take account of your latest]
No, that’s not true either. The British Museum made no assumption about the age of the cloth. They did assume that all the samples came from the same piece of cloth, and that was all.