When to "wind-your-own" vs "scavenge" a transformer?
Yes you did!
NAB=Vt for any core saturated topology including all 50/60 Hz transformers, and
Vrms=4.44fNAB for sinusodial waves. You can assume 1 for B for laminated iron core or perhaps 0.8 for a conservateive design (you are trading no-load losses for copper losses here).

Wire gauge is normal thermal heating for all low frequency transformers or rule of thumb 4 A/mm^2 for you most conservative and very long life time even at high ambient temperatures up to about 10 A/mm^2 for very small with good cooling and only ambient temperature. 

If you have pulse applications, core saturation equations are the same but wire gauge can be heavily compromised.