[name_m]Fox[/name_m]-
Male: Fennec as in the fennec fox which lives in the Saharan desert. The name comes from a berber word meaning “fox”; [name_u]Lorne[/name_u]; Bassarion is Greek for “fox”, could also be spelled Bessarion;
Female: [name_f]Lorna[/name_f]
[name_f]Snow[/name_f] Leopard/ [name_m]Wolf[/name_m]- a Slavic male name of uncertain meaning though some meanings attributed to it are “wolf”, “short”, or “snow leopard”
Snake-
Male: Python is the name of a giant snake in Greek mythology though in some versions he is depicted as a dragon, so you could use it for either one; Pytho and Pythia are female forms of the name; [name_m]Phineas[/name_m] (a Hebrew name of uncertain origin though one of the possible meanings attributed to it is “serpent’s mouth”; Viper;
Female: Adalind; [name_f]Tanis[/name_f]/ [name_f]Tanith[/name_f]
[name_m]Wolf[/name_m]:
Male: [name_m]Ralph[/name_m]; Lykos (from Greek meaning wolf"); Autolycus/Autolykus (also from Ancient Greek meaning " very wolf" or “the wolf itself”; in Greek mythology, Autolycus is the son of the god [name_m]Hermes[/name_m] and was a cunning and artful thief; he was also the maternal grandfather of [name_m]Odysseus[/name_m]); [name_m]Lyall[/name_m] (from Old Norse); [name_u]Rudy[/name_u] (short for [name_m]Rudolf[/name_m] or its feminine form Rudolpa so it could be used for either gender); [name_m]Dacian[/name_m] (the name of an ancient kingdom, the etymology behind it is uncertain though one of the possible meanings attributed to it is “wolf”); [name_m]Randall[/name_m] “[name_u]Randy[/name_u]”; Sandulf “[name_u]Sandy[/name_u]”; [name_m]Beowulf[/name_m]
Female: Ylva, [name_u]Rudy[/name_u], [name_f]Dacia[/name_f]/Daciana (the name of an ancient kingdom, the eytmology of the name is uncertain though one of the possible meanings attributed to it is “wolf”)
Dog:
Male: [name_m]Caleb[/name_m], [name_m]Colin[/name_m] (it’s both a short form of [name_m]Nicholas[/name_m] as well as the anglicized form of Gaelic [name_m]Cailean[/name_m] meaning “puppy” or “whelp”;
Female: [name_f]Colina[/name_f] (feminine form of [name_m]Colin[/name_m]);
[name_m]Bear[/name_m]:
Male: [name_m]Dov[/name_m] (Hebrew meaning bear); [name_m]Humphrey[/name_m]; Artyom (Russian form of [name_m]Arthur[/name_m]); [name_m]Orsino[/name_m] (Italian for bear); [name_m]Arkady[/name_m]; [name_m]Bernard[/name_m]/[name_u]Bernie[/name_u]; Artigal (a name with possible meanings including bear-fury or bear-ardor; Hunbald (Germanic brave bear cub);
Female: Urusla, [name_f]Ursa[/name_f], [name_f]Orsola[/name_f] (Italian form of [name_f]Ursula[/name_f]), [name_f]Bernadette[/name_f]; Artio (the name of a Gaulish goddess of bears also meaning bear, though I can see this as a unisex name); [name_u]Artemis[/name_u] (the Greek goddess of the hunt, her name is of uncertain meaning and origin though it could possibly be related to the same root word meaning bear. [name_u]Artemis[/name_u] has also been linked to bears);
Stag/Deer:
Male: [name_m]Oscar[/name_m] is an Irish name meaning “deer friend”; [name_m]Oisin[/name_m] (pr. oh-sheen) is another Irish name meaning “little deer”; [name_u]Raleigh[/name_u] (one of its meanings is roe deer clearing); [name_u]Devin[/name_u] (anglicized form of Gaelic surname Ó Damháin meaning “descendant of Damhán”, the latter meaning “fawn” from Gaelic damh “stag, ox”; [name_m]Olen[/name_m] (Russian word for deer, could also be spelled [name_m]Olin[/name_m]); [name_u]Roscoe[/name_u]; [name_u]Darby[/name_u] (a unisex name); [name_m]Buck[/name_m]; [name_m]Herschel[/name_m];
Female: [name_f]Fawn[/name_f], [name_f]Roe[/name_f] (as in roe deer); [name_f]Doe[/name_f] (means female deer); [name_u]Elan[/name_u] (Welsh female name derived from a Proto-Celtic root meaning “doe, hind”- [name_f]Elain[/name_f] is another cognate of the name (though it looks like a variant spelling of [name_f]Elaine[/name_f]); [name_u]Darby[/name_u]
[name_f]Cat[/name_f]:
Male [name_m]Felinus[/name_m] (Late [name_m]Roman[/name_m] meaning cat-like);
Female: [name_f]Felina[/name_f] (feminine form of Late [name_m]Roman[/name_m]); [name_f]Kitty[/name_f] (a short form of [name_f]Katherine[/name_f] though it’s also an affectionate nickname for a baby kitten);
*There’s not a lot of names menaings cat that I could find but since cats and lions are felines, you could use a lion name.
Panther- [name_m]Fahd[/name_m] (Arabic for panther); [name_m]Namir[/name_m]/Nimr (an Arabic word that actually refers to the panther, tiger, and leopard)
Rabbit:
Male: [name_m]Warren[/name_m] (a word referring to a colony of rabbit burrows); [name_m]Harlan[/name_m]
Dragon:
Male: [name_m]Drake[/name_m] (means dragon, snake, serpent); Apep (Ancient Egyptian god depicted as a giant snake and considered the lord of chaos. It possibly means “to slither” though its true meaning is unknown); Pendragon; Ophichus; Longwei (Chinese name meaning dragon greatness); Eltanin; [name_m]Ryu[/name_m]
Female:[name_f]Belinda[/name_f] is a name of uncertain meaning though the second element has been linked to Germanic lind “serpent, dragon, snake” or lindi “soft, tender, or linden tree”; Tiamet
Nemean lion:
Male: [name_u]Lionel[/name_u], [name_m]Leander[/name_m], [name_m]Simba[/name_m], [name_m]Leopold[/name_m], Asad, [name_m]Leon[/name_m], Lavi, [name_m]Apollo[/name_m], Arioch, [name_m]Laith[/name_m], [name_m]Levon[/name_m], [name_m]Aharon[/name_m], [name_m]Leoncio[/name_m],
Female: [name_f]Leona[/name_f], [name_f]Leonie[/name_f], Lionella, [name_f]Leandra[/name_f], Lyonesse/Lioness, [name_f]Leola[/name_f],
[name_u]Phoenix[/name_u]: the only thing I could find was Caifeng, a female Chinese name meaning “colorful phoenix” or Xifeng “flourishing/ glorious phoenix”
*That’s all I could manage to find. I don’t know if there’s a lot of names with the meanings of the mythological characters you’re looking for.