It's okay to cry. Nothing to be ashamed of. You can teach your kids that it is okay to feel something when you are sad, or to feel other emotions too. SI is one way we try to switch emotions off. If you teach your children that you have to keep your emotions back, they will also need ways to switch them off, like SI, drugs, eating disorders, ...

I am sorry about the dog. Dogs have a way of making you love them. It's what they do best. They are wonderful creatures, and should be mourned when they are gone, as any other friend or family member.