This is one of the standard arguments that is always made for not increasing the minimum wage. Raising the minimum wage might push the lowest paid workers up to a wage equivalent to other higher up the ladder, but that just means that the rest of the business is predicated on underpaying it's employees. We shouldn't be making monetary policy that placates people who want to continue to work against the interests of the economy. As with the arguments against the ACA due to "costs": If your business depends on exploitation of employees to survive, you aren't good at your business. You are on the way out anyway, so why should we continue to provide life support through social programs that you probably argue against anyway.
No, that just means you have to raise progressively all the wages that are below the mean.
Put it this way: if you just bump up Bob's hourly wage from $10 to $15 and leave Alice's untouched at $15, you have just increased the number of minimum wage workers. In a few years, $15 will become the new $10 and you will then have the same situation as today, except with more people in the lowest percentile.