Transformers have polarity, indicated by the dots on one side the windings in the schematic symbol. (pic stolen from this page)
What that means is, when measuring voltage on windings between the dot as "positive" and the non-dot as "negative", as is shown on the schematic, all the voltages will have the same sign (positive or negative depending at which point of the AC cycle the measurement is taken).
This is usually not important for power supply transformers, because we don't care about polarity of mains voltage. It can be important for signal transformers, if you want the output signal to be the same polarity as the input signal, then you have to make sure to connect the windings the correct way.
If a transformer has two secondary windings and you want to wire them in series or in parallel, then polarity is important. If they're wired wrong, in series the voltages will cancel, and in parallel they will short each other.
Anyway.
Are both wires on an AC/AC transformer live?
Your transformer has only one secondary winding. Unless one of the wires is connected to Earth (unlikely) then the concepts of "hot" "cold" "live" and "neutral" do not apply. "Neutral" means low voltage relative to Earth, "Live" means high voltage relative to Earth, with the implication that dangerously high current can flow between Live and Earth if it is given a path to do so. That doesn't fit here because the secondary is insulated from Earth.
You simply have two wires with 9V AC between them, and they're both isolated from mains and Earth, ie "floating".
Assuming it is live/hot, is there no risk in sharing ground across DC and AC components.
Since your transformer is floating, you get to choose what the potential of one of the output wires is relative to your circuit ground. Since you're using a half bridge rectifier, and connect one of the secondary poles to ground, then you made that choice. Note the transformer doesn't care, you could use any of the two poles for this.
There would be a risk if you used another circuit powered by the same transformer, and that circuit chose to connect the other pole of the transformer's secondary to ground, which would create a short. This will happen if the circuit has a full bridge rectifier, which connects each side of the secondary to ground in turn on each half cycle.
Otherwise, it's fine.
The 1µF cap should be non polarized, and probably can be removed anyway.
The 120µF cap is most likely too small for a half bridge rectifier.
Polarity of the AC measured at the "voltage sample" point relative to actual mains depends on the first paragraph of this answer.