I have an interesting problem. Water is draining out of my disposal side of the sink just fine. However, when water is stored in the non-disposal side of the sink by using the drain stopper and then released it backs up into the disposal side of the sink instead of dropping straight down into the drain, which hasn't happened before. Then when the disposal is turned on water comes pouring out of the dishwasher inlet. I have taken apart the drains and there is no backup. I ran 15' of snake down the drain with no resistance. I'm completely lost.

I have taken apart the drains and there is no backup. Well, there's a backup someplace or you wouldn't be getting water backing up out of your tubs and air gap. Did you remove the trap and snake the horizontal drain inside the wall? (see image). If not do so. If you've done that then the clog might be farther on down the line. You can get to that through the roof vent. Put out enough cable to get to the base and 18 feet more. Good luck, Tom