Re: Need some advice
It might be a good idea if you simply stated to your mom that you believe your siblings are not in the best of situations as it is now and that you are willing to take them, if in fact you are. Your mom might refuse at first, but at least she will know that possibility exists. As she thinks about it, maybe she will soften to the idea.
As far as your sister saying she is happy, she is more likely defending your mother and it would be very difficult for her to state that she would want to come and live with you.
But, again, once the idea of them coming to live with you is on the table, it's salt in the soup. or a seed that's been sewn. Nothing is quite the same afterwards - but it could take some time for anything to happen.