What did you use before you tried the mug/soap/brush?

I assume you bought your Boars Hair brush new... did you wash it with anything to get the "smell" out that new boars brushes tend to have? If so, are you sure you rinsed it well enough to get whatever out?

Suppose you whip up a big pile of lather in your mug and gently spread it on your wet face with your fingertips... just like you would spread an aerosol foam lather. Does that burn?

As thebigspendur said, Williams and Proraso soaps are pretty benign. Proraso creme has a little methol in it if I recall. Are you allergic to any soaps?