How can I explain my problems with their father to my young children in a way that does not negatively affect their psyche and morals, taking into account that we are separated without divorce due to his neglect, betrayal, lying, and theft?

There is no benefit in informing children about their parents' problems, especially while they are young, to avoid negatively impacting their psyche. General reasons can be mentioned, such as "circumstances preventing compatibility." If the children grow older, they will understand the nature of affairs. Perhaps Allah will rectify the situation between the spouses as long as no divorce has occurred.