In the current scheme, the same flow only taxes the money once: at the employer through withholding.
While the FairTax does collect tax from different entities at different points, and some tax from money that escaped taxation in the prior current scheme, other monies, now taxed, escape. It is certainly not clear that the the tax base is significantly larger, but it is different.
In this particular example, yes. But every item or service is going to see a variable effect, depending on what happens to it before it's consumed. Items with many stages of production, such as computers or cars, will realize a greater price drop, while pure services such as the nanny will not. There are estimated effects for different industries on the FairTax site.