Long question regarding children?

Your ex-wife is obviously not a good influence on your sons, so they are where they should be. You could talk to the boys, find out how they feel about their mother, and whether they would like to have a picture of her somewhere in the house. Depending on their ages, they might not think it's that big of a deal whether there's a pic or not, but you won't know if you don't discuss it with them. Alternatively you could give each child a small photo of their mother to keep in his drawer or somewhere, so he can look at her whenever he wants to, without having the picture displayed in the house.