Op, you seem to be missing the fact that when somebody dies, the things (belongings, property etc but also their debts - mortgage, credit card bills etc) they leave behind form their estate. In order for somebody to benefit from the estate, it needs to be sorted out so that it can all be wrapped up.
If the deceased had left a Will, they would have left instructions as to how they wanted the sorting out to be done and who they wanted to do it - those people are called executors.
When no Will has been left, there are no Executors to sort everything out. Therefore a friend or relative needs to formally volunteer to do it - they apply and if accepted, they become the Administrator of the estate. And they have to sort everything out. They have to pay all the bills, gather in all the money and then distribute the remainder to the people that inherit it.
It sounds like your sil has stepped up and become the administrator so that means she is the one that gets to make all the decisions about how the estate gets wrapped up. - including decisions about what happens to the house.
Although the administrator has to distribute the money according to the laws of intestacy, so sil couldn’t pay the money to herself, but she could potentially choose the estate agent that is her friend who gets a high fee and puts it on the market at a low price to some developer friends who get a great deal and who build her a conservatory as a thank you. Or maybe she does do it all properly. But if she has volunteered to be the administrator and you haven’t, then she is the one that gets to make that choice, not you.
If you want to be the one that makes the choice you need to be administrator so you need to try to apply or take over from your sil.
You still haven’t said how you expected the process to happen between your ex dying and you getting money - or as you’ve now said the house getting rented out so you could have rent money to look after the kids - somebody needs to make these things happen. How were you expecting them to happen? You need to tell us so we can help you understand what is happening.
In a separate note - if you’re a student it would be asking the Student’s Union if they can provide any access to legal support...