Doctor73, as a fellow person who considers themselves religious, let me first tell you that when I was younger I SWORE I was going to be a vet and that under no circumstances would I euthanize an animal. I went through some of the same thinking as you seem to be going through: is it playing God to decide if an animal should die and to subsequently 'cause' their death? I want to be a vet, thus I want to help animals, not hurt them. Clearly killing an animal is hurting them... or is it?
Here's the problem, I think if I truly believed humanely euthanizing an animal (for valid reasons) was playing God, then I'd also have to believe treating an animal for any kind of an illness is playing God also (as other people have suggested on this board). If God chooses when an animal should die, does God also choose when an animal should get sick? And if He does, should we go against His plan by treating the animal? It seems clear to me now that this line of thinking would result in me not becoming a vet at all.
Additionally, I've seen those cases like Bill described where you can clearly tell the animal is suffering. There's a good chance the medicine we've given to these animals up to this point has allowed them to live to be this old, and it seems to me that now we have to take responsibility for them. And despite my strong objections when I was younger against euthanasia, I now, after having given it a ton of thought, think it absolutely is the right thing to do in some situations.
Now do those situations include convenience euthanasias? I think not, and I still plan not to do those when I'm a vet. What about a dog that is scheduled to be put down anyways, but instead a student is allowed to perform a surgery on the animal first and euthanize after the student has learned something? I think this is a personal decision, and one I'm still thinking about, to be honest. And what about when an animal is truly suffering and we can't do anything to relieve its suffering except euthanize? All I know is when I die, I want to die quickly. I don't think it would be worth it to me to live 3 months longer if that 3 months was spent in excruciating pain. Do I know for sure that the animals I might euthanize feel the same way? No. But I also don't know for sure the animal would elect to be kept alive. That's part of what it is to be a vet -- to use your knowledge and experience to make decisions for animals who can't tell you what they're feeling/what they want.
Clearly, this is a topic that raises many people's eyebrows who are in the veterinary career if it's brought up. But here's the thing. Don't decide you'll do euthanasias because everyone in the field says it's okay. That said, I'm inclined to believe euthanasia is a (albeit difficult) part of being a vet. Therefore, I would do more thinking about euthanasia and your reasons for being against it. And not just because you'll have to have a heck of an amazing answer for why you're against it when you're talking to vet schools, but because you may not want to be a vet if you'll have to deal with this conversation and with seeing animals euthanized (even if you don't perform them yourself) on a daily basis.
On one last final note to my terribly lengthy response: I don't think I'll ever be happy after I euthanize an animal. I will always be sad that the animal could not be treated back to full health, and I will always be sad for the family who has to deal with the death of their beloved pet. However, I will be grateful that I have been able to help an animal stop feeling pain, and that I have allowed a family to begin the healing process instead of forcing them to watch their pet die a slow and painful death.
Good luck on your continued soul searching!