Currently, there is no way to know where your pet is in the queue(s). When you look at the nomination page, it will only tell you if the pet is nominated or not. If a pet is rejected, it will simply no longer be nominated.
A pet spotlight revamp is in discussion, and part of that discussion will change the way the pet spotlight is judged, and notifications for rejection are supposed to be added. However, no one is really sure when that will happen or what that will entail.
Indeed it is! Often, you kind of just have to guess if you are in the winner's queue, since it can take a long time for a pet to win after being nominated (Jadeyn won yesterday, but was nominated back in March). This is a combination of the spotlight only being three days a week, and the fact that the nomination queue is not checked regularly (as in, judging occurs only when needed). If your pet has been nominated for more than a month, it has good chances of being in the winner's queue... but until judging occurs on a set schedule, there is no way of knowing for sure (I've heard stories of pets being nominated for 2 months and then being rejected). But all of that is being worked on for the revamp. :)
Plus, it just makes it all the more surprising for when your pet finally wins. XD