The ballast you cant damage for sure. The 60 Hz will cause lower current in it (so underpowering the lamp) and nothing else
PFC capacitors (if installed) will be a bit overdriven (at higher current) and actually "over correct" the power factor, creating bad power factor again in the other direction (leading), ideally you would want lower value capacitors for 60 Hz for proper PFC. So long as the existing capacitors are rated for 60 Hz thats ok for the capacitor (then you only have the bad power factor but it is not a cause for damage). If not rated for 60 Hz actually i still dont expect anything to happen, but there is no guarantee since you do run the capacitor out of spec "to the bad side". If unsure disconnect the PFC capacitor
Lamp can be damaged if run out of spec at too high current (that won't happen in our case), or if run at way too low current for long periods (minor underpowering is not a problem)
Here is a quick multimeter guide for you :

And what mistakes can blow up the multimeter and should be avoided :
- In most multimeters the red wire is moved to another socket to measure current (in the 10A range or in all ranges depending on model). Be sure that the wire is pluged in the right socket on the multimeter before measuring current on voltage
- Do not measure current across terminals of voltage source - you are making short circuit (sounds like you allready tried this one)
- Do not measure current higher than the range selected on the multimeter
- The test wires of most multimneters are thin - Do not measure currents of few A and more for too long as the wires can heat up and melt the isolation
- Beware of voltages higher than the multimeter is rated for - Besides blowing the multimeter, the isolation of test wires is not rated for it either and you can get a shock. In lighting circuits such voltage is present in HPS and MH lamps which have ignitor
- Do not measure resistance in circuit with voltage present (from power source, battery, charged capacitors etc)
Update for your edit :
The converter is an autotransformer (transformer with part of the coil shared between both primary and secondary sides), but the basics for it are same as a tramsformer, so the rules of "normal" transformers mostly hold for it as well
The transformer factor is defined as the factor between number of turns in the coils. Ex. IF the transformer have 80 turns primary and 160 turns secondary then its factor is 1:2
The voltage will be scaled by the same factor 120v --> 1:2 transformer --> 240v (more turns in coil make higher voltage for the same magnetic field)
The current is scaled by the same factor in reverse 3A --> 1:2 transformer --> 1.5A (more turns in coil make the current divide by the amount of turns for the same magnetic field, so lower current)
This is consistent with the role of the transformer : It converts power but it does not create power (no fre infinite electricity....) and (assuming it have no losses) it does not destroy power either. In our example the transformer is indeed getting 120v*3A = 360VA in and 240v*1.5A = 360VA out
This is consistent with what you get there, in fact you get numbers a bit off since the transofrmer does have losses so it takes a bit more power on the input which is wasted as heat