Regarding the Squirrel cage mixer:
Do NOT use galvanized metal in the presence of strong acids or bases (i.e. NaOH). The coating may dissolve, and the zinc is toxic. This is more of a concern with food, but don't take chances. Your skin will readily absorb chemicals.
Stainless steel (food grade), and HDPE are the only materials that should come into contact with your soap. There are lower grades of stainless that are not safe for use in soapmaking. Be sure you are using stainless intended for food preparation. Regular ferrous steel should not be used.
There are many grades of plastic that are not safe, although HDPE is not the only one that is safe, it is certainly the most reliable. Buna is OK, and of course there are other plastics intended for food production. Do the research first.
Aluminum is really not safe.
Wood is OK, but may leech tannins. Tannins are not really a concern for anything other than discoloration, though. It's fine for molds if you line them. Make sure the wood is not finished, though.
And on and on... Just stick with quality stainless and HDPE and you should be fine. If you are not sure, don't use it.
The point is that if you're going to use a mixer intended for paint, you should be sure it is made of food-grade stainless.