In the static case (left image) this is in essence the same as a mercury barometer, only with water. The column would be about 10 meters high to reach close to 0 Pa at the very top. There, a gas would form: water vapour in equilibrium with the liquids phase below, i.e. the gas at vapour pressure of water (approx. 30 mbar=3000Pa at 25C).
The dynamic case reminds me of a Sprengel pump, where falling mercury creates a vacuum above. https://en.wikipedia.org/wiki/Sprengel_pump
Q1: What is the pressure distribution before the seal is broken? (left picture)
Answer 1) If the vessel was filled at ambient pressure completely and you have a stiff container, the pressure inside will be one bar for the liquid. Above a vapour phase will form at the vapour pressure of water at the given temperature for 1bar. Going down the column the pressure rises with depth linearly.
Q2: What is the pressure after the seal is broken?
Answer 2: After the seal is broken, water, initially at rest, will be accelerated, by a large force due to the high hydrostatic pressure. This will quickly increase the gas volume at the top, hence decrease the pressure in this part of the vessel. As a consequence the water at the top could start to boil even at room temperature, filling the vacuum with water vapour.
Q3: Can the pressure go negative given a long enough hose?
No. The pressure might go below vapour pressure while the vessel empties. This will depend on whether the water moves out alone or bubbles can enter to travel up, causing friction and letting water out only slowly.
In a really long column though, the opening of the valve causes a huge pressure drop, which might travel upwards the column (with speed of sound) and hit the top very hard. This effect is known as water hammer in pipes when valves are opened and closed: https://en.wikipedia.org/wiki/Water_hammer.
Q4: Will the barrel implode violently?
Answer 4: Unlikely from static pressure. The load on the vessel at the top is maximal 1 bar from the outside. If it withstands this in the beginning, it should be fine later also.
Dynamic effects like to one above can become important for very long tubes. But they would not cause an implosion, rather an explosion.