…How does Vzrc (zero rate climb) and Vso compare in std. atmospheric conditions at sea level for some different aircraft?...
On many aircraft types, Vzrc will be below Vso, and the aircraft will stall before it runs out of excess thrust. I suspect that this is probably true for most aircraft types.
There are some aircraft types, particularly delta winged ones, will reach Vzrc before Vso.
However, unless we talk about a specific aircraft, it is hard to comment further.
…I understand that the Concorde has published Vzrc speeds…
…Is Vzrc only relevant with an engine or two out?...
Vzrc is always relevant, in the same way that a stall speed is relevant, in determining the minimum safe operational speed of an aircraft; however, there is a significant difference. Broadly speaking, Vso doesn’t increase following an engine failure or two, but Vzrc certainly does.
So Vzrc becomes a lot more
relevant when the total thrust available is reduced following an engine failure.
When we talk about Vzrc, we are talking about a rate of climb (even though it is zero), and an aircraft climbs because of an excess of thrust (not lift). The number of operating engines, and hence the total amount of available thrust, is critical to determining that speed.
…Why is it that even though the wing isn't stalled and the Vzrc speed is well above stall-speed that the aircraft can't climb?...
There is a very good answer from FredT
I would just add that he is referring to slowing down below Vmd, the minimum drag speed, when total drag also increases as the induced drag increases.
…This seems to contradict information I've read in other threads….
I’m afraid that’s the story of aviation!
You will have to make your own mind up about which threads, sources and posters you should believe.