So there are several issues here. For the backdrafting, it could be an obstruction in the vent between the water heater and the outside. It could be a furnace near the water heater that sucks air out of the room when it comes on. Or it could be that the vent doesn’t go straight up, but goes horizontal for some distance before passing out the roof or a wall. That last is where a power-vent heater would be needed.
For the not so hot water, the fellow was talking about a cross connection. You can read about that and how to test for it here: http://www.waterheaterrescue.com/Troubleshooting/Tanklets/cross-connections.html . People usually will swear that nothing has changed and it’s impossible for it to be this, but not uncommonly, something has changed that they don’t know about.
For the high water bill, read this one: http://www.waterheaterrescue.com/Troubleshooting/Tanklets/high-water-bills.html . With luck, it will be something that’s not too hard to get to.