Well I hope you don't think I give a better appearance through artificial means - I assure you every detail is 100% me! (Though perhaps I could do with a little gingering up)
I think now you come to mention it, I have come across 'figging' as an old word for 'gingering up' horses (and of course that's the origin of that phrase too), but like most slang words the origin is obscure. Dutch vegen, German fegen, as you say, means 'sweep, polish, prink up', so might have shifted to mean making something look better than it really is (the root is 'fag-' as in Old English fager, modern 'fair', which figging ain't). But 'feague' in English, probably from that same Germanic word, had earlier meant 'beat up, sweep the floor with someone, polish them off', in rather a different sense! And then there's also the 'figge of Spain' (la figa), the rudely contemptuous gesture which the horse-trader might well give to the departing customer he's just fooled into buying a worn-out nag (hence 'not caring a fig').