Does personality prove self-consciousness? If an animal has a defined and changeable personality (can get depressed, etc.), is it then self-aware?

There is a difference between consciousness and sel-consciousness. It is one thing to be aware of things around oneself, or even of one's own physical states, and another to be aware that one is so aware. The latter is self-consciousness. Personality is yet a third thing. As a reasonable first pass we could take it to be a bundle of behavioural, affective and cognitive dispositions. Those would seem to require consciousness, but not self-consciousness.

I have a 12 year old dog. She's no longer in great health, doesn't qualify as cute or attractive, and has rightfully been accused of stinking up any room she remains in for more than a few minutes. Still, she's my dog and I love her. Unfortunately, I am in a situation that requires that I move to a place that won't allow me to bring her. I can't find anyone to take her and am pretty sure that if I take her to the animal shelter she will spend a terrible 2 weeks there, not be adopted and then be euthanized. I've been thinking of taking her to a veterinarian who will put her to sleep with a painless injection while I'm there with her. I know this will break my heart, but is it the right thing to do?

I agree that this is a hard one, harder of course for you than it is for us, and harder for us than it is for your dog. The last is an important point. Euthanising your dog in this situation will cause you a great deal of suffering, but will not cause the dog suffering. It will deprive her of future pleasure of course, but she will not suffer from knowing that. I think that condemning her to spending her last few weeks without you in a cage would be cruel, and recommend against that course of action. Of course, if you can, you might arrange a private adoption. But if you can't, I think that euthanasia is the most humane option. You can ease your own feelings of guilt by remembering that. But you probably can't do much about the reasonable grief.