Think of it this way: "rely" is almost like "depend" the difference is that "rely" is often used in the sense of a person relying on something or someone. "Depend on" can be used the same way, but when discussing how one event can't happen without another, we are more likely to say "B depends on A" than "B relies on A".
However, "hinge" is taken from the word for the hardware a door swings on, and is extended to a metaphorical sense that is related. So the best place to use "hinge" in this metaphorical sense is not when B's existence or nature depends on A, but rather when the direction that B can go depends on and is constrained by A. And not physically. Often historical events are said to have "hinged" on some condition. If the condition had been otherwise, the outcome would have, figuratively, "swung" the other way. So if you say that event B "hinged" on condition A, it means that A somehow guided the outcome (B).
And I don't know of any case where we would say a person "hinges" on anything, or that anything "hinges" on a person; but a person's actions, or success, etc. could "hinge" on someone else's actions.
You wouldn't say, for example, as Avigrail seemed to imply, that a baby "hinges" on its parents; it "depends" on them. It is unlikely a baby knows what it means to "rely" on them. But you could say the baby's eventual character "hinges" on how his parents discipline him.
Nor would you say "we can't hinge on him to tell the truth", we would say we can't {rely on, depend on, count on, trust} him to tell the truth.
So, I suggest that you would not be speaking idiomatic English if you were to substitute "hinge" in ANY of the examples you cite.
I hope this helps; I wish I could cite some source of more examples where "hinge" is appropriately used.