I've never understood these practices either.  They should use the faldstool and dispense with the crozier (not sure about the mitre).

The mitre is proper to all episcopal and abbatial liturgies, existing in its three forms:  simplex, auriphrygiata, and pretiosa. There first two are used during Mass. The latter is used for processions, blessings, audiences, etc.

And, yes, all trad bishops should be using the faldstool for Mass since none are ordinaries in their own jurisdiction.

The SSPX bishops also have the strange practice of an pontifical Missa Cantata. Whilst such a ceremony exists for the Novus Ordo, episcopal Masses in the classical Roman Rite do not admit any such Mass. There is pontifical Low Mass, pontifical Mass at the Faldstool, and pontifical Mass at the Throne. But there is no pontifical Missa Cantata.