Workings of a microwave oven magnetron transformer

Since transformers by their nature are bi-directional, the selection of the primary side totally depends on your input voltage and desired output voltage.

The transformer you describe likely has multiple taps on the "primary" side, may have multiple windings on the "primary" side and likely has multiple windings on the secondary side. Start with a low range DMM, and check for continuity between different leads on each side of the transformer. Once you have mapped continuity, check resistance between the same leads. You should be prepared for the transformer to be as complex as this:

The "secondary" side may be a single coil with multiple taps, or it may have multiple outputs more like the above example.

Once you've reverse-engineered the coil arrangement, you'll need to determine the turns ratio between each set of coils. I would NOT recommend your 120VAC test for this. Start with a much lower (and safer) voltage. Find a small "wall-wart" type power supply that you can sacrifice. The lower the output voltage the better. You want it just for its transformer, not the rectification and regulation components, so if you can find an AC-output wall-wart, you can use it's output as-is. What you want is a low voltage AC source that you can use to test individual windings. Note that applying a low voltage AC source to the "secondary" may result in lethal voltages on the "primary", so be careful!

Find one set of windings to apply your AC input to, and measure the resulting output on each set of coils and on each tap. Transformers are ratiometric, so the relative voltages will be the same using your low voltage AC test vs. when you identify the intended primary winding and apply 115VAC to it.

Doing this, you should have a good sense as to what windings are present that the relative turns ratio between each. Good luck!

You can use a directional coupler to measure VSWR (reflected power). There are many app notes online for this: http://www.minicircuits.com/app/COUP7-2.pdf here is an example.

However, the fact is that waveguides need to be tuned because no two are exactly the same when you put multiple components together. This means you won't be able to just change the height a set amount to 'fix' the match. You could characterize each component and get a set of "heights" to adjust, but I would recommend trying to have a control loop where you are constantly measuring the reflected wave and tuning to minimize power.

And again, the VSWR is kind of a derived term and it will be best thought of as "tuning for minimal reflected power" when setting your system up.

And wowzers, I just saw that you have a WR340. Those are gonna be some hefty stubs!

edit: Do you really expect your match to be that dynamic? If not, you can just tune your waveguide piece on a network analyzer, epoxy/lock washer the screws and you'll have yourself a well matched piece.