The sensory bonus and talk-to-animal-types are generally relatively straightforward, so here's my take on the normally-impossible-thing / limited skill bonus.
Viper: Ability to "see" heat signatures; in terms of game mechanics this is mostly being able to see in the dark, though a creative player might use this as an excuse to tag their high concept on a wider variety of things - for example, an empathy check for reading someone's emotions... (Note: this may also depend on the kind of viper; not all viper species have this particular sense.)
Python: +1 might when initiating or maintaining a grapple; simple, straightforward, and to the point.
Wolf: Not sure here; something to do with teamwork, maybe?
Housecat: Add an escape artist trapping to athletics - mostly useful for squeezing into/through narrow places where people would swear you couldn't fit...
Spider: Depends heavily on the type of spider. Could vary from a craftsmanship trapping to allow creation of webbing traps, to +1 athletics when jumping, to +1 stealth when not moving...
Outsider: Add an unreality trapping to conviction - you can make maneuvers that establish aspects that emphasize the breakdown of reality in your presence. Things like "Lurking Shadows" or "Are those mice chanting?" or "Strobing Octarine Light"...
Shark: Depends on the type of shark; the common option would be a truly extraordinary sense of smell - when you're in the water. But I think the most fun option would be the electrical senses of, for an example, the hammerhead shark. You'd be able to sense the presence / absence of living things, even around corners, tell the difference between a normal person and a lobotomized renfield, etc.
Turtle: Depends on kind of turtle. For a relatively ordinary, say, box turtle, you might get +1 to defense against physical damage from edged attacks (but not hammers, bullets, etc). For a desert tortoise, +1 on endurance when dealing with heat / lack of water. For a sea turtle, +1 athletics when swimming. Etc.
Crab: Really no clue. I am sadly not particularly up to date on the special abilities of crabs.
Bat: Depends on type of bat. Echolocation is the obvious thing for the types of bats that can do that. However, there are a number of species (for example, fruitbats) that rely on sight and smell rather than echolocation; for those... hm. Nothing immediately springs to mind, sorry.
Bear: +1 intimidation when you're bigger than your target, maybe? Or you could go with some kind of athletics bonus; bears are surprisingly fast and agile for their size.
Horse: +1 athletics when sprinting.
Elephant: I'm tempted to say "extra minor physical consequence slot", but that's probably too strong. Maybe something representing the elephant's ability to be really really loud?
Falcon: Ability to see stuff in perfect detail at extreme distances. Or possibly just a +1 athletics when sprinting.