1. All one-way valves depend on flow. If there is a higher pressure from A to B. And that's the direction of resistance, then in a valve with moving parts, a ball or diaphragm of some sort would be pushed against a lip, and the flow would stop. That still depends on a pressure difference where flow was traveling from some high pressure point to some low pressure point. Let's say that valve was off design for the flow rate. It may not have enough pressure to seal, and it would still be leaky. The Tesla Valve's performance is measured by its diodicity, and it has to be matched to the conditions it's supposed to exist in. If it was really well designed, it's diodicity would be very high and thus leakage would be minimized. From an engineering standpoint it still usually makes sense to use valves with moving parts on your example lox tank. As the main way of checking flow, they are well understood and very reliable. However, if you are in a situation where a moving part is a negative, like in microfluidics, a Tesla valve might make more sense.
2. Liquids and gases have different viscosity, so the Tesla Valve would be different for each fluid and flow regime. I am unfamiliar with this series-of-petals geometry you are remembering. However, it reminds me of this excellent work done on optimizing topology for different Reynolds numbers in Tesla Valves: http://www.senlin41.org/topology-optimization-of-tesla-type-...
3. One of the original intents of the valve was for Tesla's Bladeless Disc Turbine. In Tesla's time, materials were not what they are now, and thus valves with moving parts were not as reliable. Also, getting a valve that can close and open with high frequency is not always easy. The valvular conduit that Tesla designed was his solution to this problem.
Thanks for the book reference. I read "Wizard: The Life and Times of Nikola Tesla" But I hadn't seen this other book. I might pick it up.