If you have an animal companion, which I've never had one, does the companion roll it's own initiative and act on its turn in the sequence of events?
The most common practice is that it goes with its humanoid, just for ease of use and to save the bother of figuring out what happens when you tell it to attack something that the fighter killed in between turns.
On the stats that I've found, does the multiattack feat factored in already on Cat attacks?
-Multi attack only applies to secondary attacks. This was a lot more useful back in 3.5 when companions actually had secondary attacks. Most of what they have now are primary attacks, so multiattack is generally useless.
Does anyone have a major listing of animal companions and their stats?
I'm meaning like a link to somebody's spread sheet or something that points out the best type to grab?
Thinking of using the new Hunter class or Inquisitor archetype the Sacred Huntsmaster.
So far I'm leaning towards a cat for their number of attacks, but a wolf wouldn't be bad for the trip/scent tracking stuff.
-Companions are generally good enough that any of them will be pretty useful. Cats are definitely up there: pounce is amazing at higher levels. You can easily out damage a fighter in very mobile fights.
Your best bet for how to handle your animal companion's initiative is to ask your GM, "How do you want to handle initiative?"
As BNW mentioned, the most common method is having your companion act on your initiative, but I've also frequently seen GMs have initiative rolls for both the PC and Animal Companion (especially if the PC has a significantly higher initiative modifier.) In the case of the later, the Animal Companion delays and "Guards" until given a different command.