What are some saints that have stories about them (and icons) that have animals playing a major part?
Off the top of my head:
Greatmartyr Mamas and his lion
St Gerasimus of the Jordan and his lion
Various warrior-saints on horseback, such as George, Demetrius, and the two Theodores (the General and the Recruit)
St Mary of Egypt, and the lion which dug her grave
St John of Shanghai and San Francisco and his dove
St Helen, mother of St Constantine, who brought a shipment of cats to a snake-riddled region in Cyprus, and the monastery dedicated to St Nicholas of Myra which was established to honor this gesture; the monastery still exists, and is known as St Nicholas of the Cats (Aghios Nikolaos ton Ghaton
St Sergius of Radonezh surrounded by various critters in the wooded wilderness where he set up his monastery
St Seraphim of Sarov and his bear
.... and the innumerable monastics, sainted and otherwise, who have had cats as their companions. One monk I know and love dearly once told me: Cats are practically a canonical requirement if you're a monastic.
He wasn't completely joking.