How many guns do you need simultaneously? Is the number 10?
I'd start with an RV water pump. They are 12V and made to maintain pressure whether water is flowing or not. I once used one for a custom portable water sprayer and used two pressure washer wants with triggers on each one and could get some pretty serious flow with each connected through 30' of hose. When we adjusted the nozzles to shoot a stream, it was WAY more than you'd need for this game, so I suspect one RV pump could support a good many guns. At least five, and maybe up to ten. I'd bet you don't need more than two. They do take a good bit of current, though! So an option for power is to just use a decent sized car battery and just charge it between uses. Otherwise, 12V DC power supplies that can supply a LOT of amps can be expensive. You'll have to check specs to see how you want to go on that front.
The next best option is to just use a well water expansion tank. You won't need a huge pump, and in fact can probably use a single RV pump for this. It'll basically act as a big baffle and the pump will have plenty of time to recharge between races. You'll find these at Lowes. Not too large physically.
I see no problem whatsoever with your water column filling idea. It's not THAT complicated. I think with decent plumbing and the same sized valves that you don't have to sense your water level, just base the "winner" on cumulative time the valve is open and call it good.
I'd definitely go with the small plate on a spring loaded switch for the target. It'll just be annoying to make the switch waterproof-ish.
There are a ton of relay boards on eBay that are perfect for driving your valves.
If I were doing it, I'd make each gun have a coin slot and automate the money part, too. But I grew up in the coin op industry and have a pile of video game coin doors laying around. I did a sophomore project in college way back that was a 3 digit slot machine (just had 3 seven-segment LEDs that rolled at different speeds for different amounts of time to a number). You could put in from 1-9 quarters and then hit start. If you had any two numbers match, you got something like 2x the quarters you put in paid back in tickets from a ticket dispenser. If you got all three, you got like 5x your money in tickets. Those ticket dispensers are pretty easy to control as you simply apply power and then count pulses and stop power when you get however many pulses you wanted to dispense in tickets. FWIW.
--Donnie