No, 'proper' refers strictly to colour. A unicorn rampant, a phoenix, a lion passant guardant can all be proper. In the case of mythical animals it's a bit dodgy of course, but everyone agrees that unicorns are whitish. Phoenixes are multicoloured with crimson or purple predominating -- a phoenix proper is used in the arms of the Painters' Company IIRC.

Only very rarely are natural-looking versions of animals used, and they're usually described as different beasts. The dolphin is a bizarre monster, as is the tiger or tyger. To specify a natural one you would say Bengal tiger or whatever: examples occur.

Metal on metal is occasionally seen in Continental heraldry, e.g. the arms of the Papacy, St Peter's keys or on a field argent.