No, you want to inject some signals without making it less stable, so you can see how it reacts and how stable it is, if you modify the characteristics of the circuit you can't know how the circuit behaves once you removed the transformer.
Floating means you apply a voltage at the terminals of the transformer, but anything you put at one end is seen at the other, as it's not necessary referenced to ground or some arbitrary point but rather to whatever is in the other end. Galvanic issolation isn't strictly necessary here, you could use an active floating signal to inject but that is harder to behave as it should without modofying the properties of the circuit.
Node is a pretty basic concept in electronics, check kirchoff laws as you really need them to make any sense of electronica. Basically, a node is a point where you connect stuff, the entire conductor connecting all it's components is a node, like a trace in a PCB.
With an injection transformer you would be pinching a node, cutting into it and introducing the signal. When the signal is not present the node behaves as usual, when you add a signal it's pressent referenced to that node if seen from the other end.
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