N.A. household is what I call "split-phase" AC, with two 120Vac signals of opposite phase coming in. Can be combined to make 240Vac for larger motors like dryers or ovens, usually used separately for regular stuff.
Personally, I wouldn't short any single coil's wires together, as I think that would waste/suck the emf from the core and cause current surges in the shorted wires. But I'm no xfmr genius, so I'm not sure
The 2 black and white wires in the photo appear to be the only wires going to one side and look like household AC gauge, though, so maybe the output is center-tapped, not the input
If it's on an input coil (may be more than 1 independent input windings) then it's used to choose between input voltages, like between 220V and 110V. If it's on an output coil then it's used to either choose the output voltage, as a 'sensing' wire for a control system (it'll be low voltage/low inductance), or(/and) to power separate parts of a circuit (e.g. analog/high voltage and digital/low voltage).
@AndyD273 nah, don't worry about background, just look at schematic of any transformer with multiple taps and coil in/out hookups. They can only get so complicated. A "tap" is just a short coil to detect the field (as opposed to driving it/input power, or receiving it/output power). Used for control systems. Cheap transformers don't have them.
Used to add a bounty to questions I thought were pretty good but with bad/no answers, usually because it was a difficult question, but I can't figure out how to do that anymore.
Can I just not find the 'add bounty' link/button that used to be on a question for those users with sufficient rep, or is it gone, or was the rep limit raised above mine? (6ksomething)
@AndyD273 you can probably figure out which order the coils are in relation to the wires just by measuring resistance with a good meter, as it'll be ~0.1-1'ish ohms. The tap wires, if there are any, won't be measurable (they're too short) without a great meter (eg. 4-wire, fancy Fluke, etc)
It sounds like you're looking for a way to test your program. Maybe look at writing a script for the VM to run to send, receive, then do something with the received audio (e.g. with a built in BT stack, it could send it to your computer/phone for further analysis).
@user17915 Probably not. Looking at a link that Google found for me, it looks like the DSP gets the audio independently from the VM, but I doubt that it takes care of the Bluetooth pairing/stack. I bet the VM does that.