1. You might be able to wind up the estate without the need to get probate. It depends where the money is held. Some banks & building societies are willing to pay out relatively small sums on a death certificate without seeing probate documents. You need to ask about this.
2. You say you moved back to live with your mother "recently". Assuming she had a normal council tenancy, you would have had to be living there as your only home for twelve months prior to her death (as well as living there when she died) in order to inherit the tenancy. Also, the tenancy would have to be one which had not been inherited before (in other words, your mother - or her jointly with someone else - must have been granted the tenancy without getting it by inheriting it from another family member). Some tenancies have more generous terms than these, so you need to check with the Council.
If you cannot inherit the tenancy, the Council will not have a duty to house you if you are of working age (under 60) and not in priority need - which means you do not have children living with you and are not vulnerable by reason of health problems.