It filled through the hole in the top that you see under the faucet when the cover is in place. That is mentioned in the picture in the lower right corner.
You could not use soap; it made too many suds. As has been discussed here previously in other threads, there were no non-sudsing surfactants until the 50s and those were not very good. Early dishwashers used a combinations of water softening agents, like STPP and others. Because they were not fully built detergents, early machine dishwashing was not like today. Food soil was quite thoroughly removed from the items before they were washed in the machine. The dishwasher with the water softeners removed the grease and food films. You basically washed in the left hand bowl and loaded the dishes into the dishwasher on the right for what amounted to hot water rinsing.
I want you to look back at that dishwasher tank. Do you have any idea how much very hot water it would take to heat that up until it did not cool the wash and rinse water? If you wanted half-way decent dishwashing, you had to run several changes of hot water into that tank to heat it before washing dishes. This is why most of these machines were almost more bother than they were worth and were mainly used for rinsing dishes that were already washed. Also remember that storage water heaters were not common in the 20s and 30s, so a house would have to have a good supply of hot water to use a dishwasher, although a home that could afford a dishwasher probably was wealthy enough to have a good supply of hot water.
It is correct that this machine had no timer. The control would have been wash/rinse or drain.
