No, I wouldn't drain a sink or washer into the floor drain. However, your water heater has a pop-off valve on it which could be piped down to within 2" of the floor; then you could occasionally pull the lever on the pop-off to fill the floor drain.
Yes, you certainly can make a floor drain deeper, Kentucky code allows a water seal of 2" to 4". The 4" water seal would cause it to take longer to evaporate. This deeper P-trap can be made up with a return bend, a short piece of pipe and an ell. The depth of a trap is determined by measuring from the top of the return bend to the bottom of the outlet pipe after the ell (non-technical terminology). You will know when to add water when you look down in the drain and don't see a full round circle of water or if you smell it.