1) From the picture it looks like your Rain Bird transformer is indeed 24V AC (I get that because it says Output: 24V ~ 50Hz: you will only indicate Hz if it’s outputting AC)
2) Since you have an existing 24V AC adapter, you can certainly get OpenSprinkler AC version.
3) OpenSprinkler AC is only compatible with 24V AC valves. OpenSprinkler DC is compatible with both 24V AC valves, as well as non-latching 12V DC valves. Neither of them is compatible with latching valves (many 9V DC valves I’ve seen are latching, so be careful). To tell non-latching from latching, you can typically measure the resistance of the two wires of the valve — latching valves have very low resistance (<10 ohm) while non-latching valves have larger resistance (20 ohm or more).