You put your two-dollar bills under your twenties. You don't usually give twenties out as change, so you won't run into the twos. And you put your dollar coins in the penny or dime slots so you won't confuse them for quarters or nickels.
WRONG!
We do so give out twenties as change. We give them as change every time a customer hands us a fifty or hundred dollar bill.
I have had customers hand me hundred dollar bills for a less than five dollar purchase FIRST THING IN THE MORNING. When I have had NO large bills in my drawer to use for making the change.
We also give out twenties as change when customers ask for cash back. They can do this every time they use a check or debit card to make their purchases, or every time they use a Discover card. Discover even advertises their card as a means for getting cash back, on national television.
As for putting the dollar coins in with the pennies or dimes, this would make it very difficult to simply slide the pennies or dimes out of the slots. These slots have curved fronts, to make it easy for the cashier to get to the coins. And the dollar coins do NOT sink to the bottom of the slot. They float on top of everything else - small items always slip under the large items (which is why farmers always find large rocks finding their way to the top when they plow their fields).
Not to mention how this would make things more difficult when the assistant floor manager comes around to audit the drawer, to make sure no errors are being made by the cashier. This process is done with a machine these days - the auditor takes the bin from the cash drawer, weighs it on the auditing machine, and gets the total value of the bin based on the weight. Mixing coins would throw this process off.
(The dollar bills are also handled by weight, btw. Fascinating process.)
While we're at it, a sidenote on the change given by the self-scan registers:
The self-scan machines do not carry ten dollar bills, to be used for change. The machines have three bins under the change slot: one for twenties, one for fives, and one for one dollar bills. It was decided that having a bin for ten dollar bills would take up too much room, and be redundant for something that can just as easily be resolved by dispensing two fivers.