Winding ratio is basically input voltage:output voltage + a fudge factor for copper and iron losses (which will be dependant on the transformer in question).
Like a 120V to 12V transformer will be about 10:1 turns ratio, plus a bit on the secondary side, so more like 10:1.1 or 10: 1.2 or so.
To be 100% sure of turns, you'll need to unwind the transformer or find a datasheet.
To test the output voltage, just disconnect the output of the transformer from your circuit and turn it on with a voltmeter actoss the secondary.
You'll find the open circuit voltage is a bit higher than under load, so you can 'back probe' the secondary connector while it is still connected to the circuit board (or another set load) to get the voltage under load.
Knowing the input voltage, and loaded (and load current) and unloaded output voltages with a little math can help you find more of the transformers parameters.