Im going to try to make an autotransformer ballast for a 400 or 1000 Mercury/ metal halide. First: does anyone know where I can find a set of calculation data for this? Second: will a microwave transformer core be sufficient. Third: what gauge of wire will I need for the winding. Thanks.

The autotransformer ballast has to combine actually two functions for a HX and four for CWA (All units are SI, so m, H, A,...).

The design of a HX will go along following equations (the CWA is way more complex to design):

- Stepping up the voltage. That is the easiest part, it means a 220..230V sectondary (to reach the required 350V OCV, when added onto the input voltage)
Nprim >= Vmains / CoreCrossSection / Bsat / 4.44 / Freq
Bsat is the saturation flux density, or better the maximum flux density you want to use for your design, could be limited by e.g. core losses chart,...
If you start from an existing ballast or transformer, you have this given.

Nsec = Nprim * (OCV-Vmains)/Vmains
This assume the secondary being hooked directly on top of the primary; if it is connected to a tab, use the tab voltage instead of the mains for the equation

The wire thickness is determined by the heat the ballast body is able to dissipate (acting as a heatsink on it's own), dictated by the maximum wire resistance. And the wire resistance is
Rwire = Length / CrossSection * 30uOhm.m
Note: At 25degC the copper resistivity is about 16uOhm.m, but you have to count for the operating temperature, that means double resistivity for 150degC.
Now you should verify, if the wire fits into the core. If not, you have to select larger core (thicker stack,...)
Now you should verify, if the

- Provide ballasting inductance. This is done by a magnetic shunt with accurate gap, the cross section is so, for the CWA it sasturates at predetermined flux, for HX it does not saturate. The gap is calculated as any other inductor, starting from the secondary turnms:
Inductance = sqrt(OCV^2 - Varc^2) / Iarc / Freq
ShuntArea >= sqrt(OCV^2 - Varc^2) / Nsec / Bsat / 4.44 / Freq
(it is the same as for any transformer core cross section, just the voltage is the virtual voltage across the inductance component and the turns represent just the secondary)
Gap = ShuntArea * 12.56E-7 / Inductance * Nsec^2

As it is usually very difficult to alter the gap size, initially use thicker stack for the shunt and it's calculation (larger Area), then adjust the current experimentally by adding/removing pieces from the shunt stack, so increasing/decreasing it's cross section area. The larger initial cross section means you have margin towards driving it into saturation (that is not wanted at all for the HX style)

I don't think the MOT would be usable. 1'st it does not offer any space for the magnetic shunt, 2'nd it has an air gap in the main magnetic flux path (as power factor compensation for the capacitive secondary load) and 3'rd the MOT is designed for forced air cooling and rather limited lifetime (so high operating temperatures; household microwave won't operate more than few 1000 hours over 10 years of use, most of that only in very short periods,, so it does not have any time to heat up much)

Could I maybe make 250 or lower wattage with it?

I'm afraid just the power factor compensation air gap would be responsible for enough primary current to overheat it without the forced blow cooling.

Plus how are you going to make the magnetic shunts? The resulting air gap would be in mm range or even smaller, so I do not know, how you want to assemble the thing...

Normally ballasts are made first with way longer center pillar, so the two coils plus the shunt could fit easily while still maintaining convenient shape (mainly the winding is not that far from a square window). The MOT shape is made to just fit two windings without any space for any shunt or so.

Don't forget, the shunt has nearly the same cross section as the main core, so it is no small thing at all. And it does not scale with power, the cross section is just the matter of voltages and number of turns and both are given...

So I'm affraid MOT won't work for a ballast...