Minno has it right. The two valves you describe are the low point drains for the hot and cold water lines. I recall these play a role in a quaint process called "winterizing" that is practiced annually by Airstreamers who do not live in Florida.
Whether you pressurize the water system by filling the fresh water tank and turning the pump on or hooking up a hose to the city water connection, if those valves are open water will flow out of them. Opening sink faucets will let air in faster and speed the flow.
The white plastic drain you describe sounds like the fresh water tank drain. On our 2014 25' EB that drain is between the tires on the STREET side... maybe on your year and size its on the curb side as you describe. If that's what it is, it drains the contents of the fresh water tank. That is also a step in winterizing, but most of us drain that tank for storage and refill for a trip.
If the low point drains were open you should have seen water flow out of them and little or no water pressure evey time you used the water regardless of whether it was from the fresh tank or a city water supply. Was the incident you described the very first time you used water with a new trailer?
As for the drain on the fresh tank, if you had not filled the tank and the drain was open it should not have drained anything when you hooked up the city water supply line. There is a check valve between the two supply sources (tank and city) that should prevent the pressure of the city feed from filling (then overflowing) the fresh water tank.