Oh Carla, how sad and how difficult for you. Don't feel bad for grieving so much when others have lost family - only you know how deeply you feel for your pets. I too had my dog put down, and held him in my arms as he died, so I understand how awfull you must be feeling.
I don't think there's any harm in explaining that the dog was so ill that she couldn't get better, and that the vet gave her a medicine to make her go to sleep for ever, so that she wouldn't be in any pain any more.
TBH I found it very distressing to see and to hold my dead pets, so I'm not too sure about showing her to them, but you know your daughters and maybe it will help them to say goodbye.
Burying her in the garden and letting the children help plant a bush, or choose a piece of sculpture is a wonderful idea. I wish we had done something similar, but the my mum did the gardening and she hated the idea.
It's alright for them to see you cry - that will allow them to cry if they want, and I'm sure will give you the strength to pull yourself together as well when your girls need you to do so.