What are my rights regarding the rental property?

First, his mother has notihng at all to do with this: while you can "ask" her for money (anyone can ask for anyone else for anything), she has no legal obligation to  you or her grandchild.
Second, any agreements, contracts, leases, etc. he has signed are ones which he is obligated on: he can be held liable for his portion of any payments. However, that doesn't change the fact that if you don't pay, you will be the one who is evicted or has utilities disconnected, etc. (For example, the landlord does not need to take your domestic situation into account: if the landlord does not receive the full rent, he can evict.) So that fact that your husband could later be held liable for all these contractual obligations will not help you in the near term.
Third, if he did not sign a written agreement for something, he is not--until you bring a divorce case (see below)--obligated to pay for it; so he does not have to pay for groceries, for example. 
There is a way, however, to hold him liable to honor his contractual obligations, to pay support for you (spousal support or alimony), and to pay child support: by divorcing him. In a divorce, you could get child and potentially spousal support awarded; get any assets (e.g. vehicles; money in the bank) divided; and get an order that he pay certain expenses. There are even ways during a divorce, to get a court order fast that he cover certain expenses on an interim basis while the case is being fully decided. You are advised to speak with a family law attorney immediately.