Alternative forms 
Earliest sense is “fart”, and later “stink”. Sense of dog is 1805 American English, from earlier forms fice and fist, as abbreviation for fysting curre “stinking cur” (1520s), from Middle English fysten, fisten (“fart, break wind”) (mid-15th century) (related to Old English fisting (“stink”)). Old English term is from Proto-Germanic *fistiz- (“a fart”), presumably from Proto-Indo-European *pezd-, though this is disputed.
One explanation for the association of farting with small dogs is given in the 1811 slang dictionary, which suggests that the dogs were blamed for farting, specifically defining fice as “a small windy escape backwards, more obvious to the nose than ears; frequently by old ladies charged on their lap-dogs.”
Cognate terms include Danish fise (“to blow, to fart”) and obsolete Middle English askefise (“bellows”, literally “fire-blower, ash-blower”), from Old Norse; originally “a term of reproach among northern nations for an unwarlike fellow who stayed at home in the chimney corner”.
feist (plural feists)
- (US, regional) a small snappy belligerent mixed-breed dog
- (vulgar) A silent (but pungent) fart (flatus)
Usage notes 
The term feist is uncommon, but the derived term feisty is common.