I didn't read all the post, so if this has already been said I am sorry. First let me say that I am so sorry for your loss and for your sister's loss. You are going through so much at this time. Know that I am praying for you through all your trials.
My Mother just passed away last month and I was involved in the entire process.
When we went to the funeral home we made sure we had the following to avoid extra trips. We had all life insurance policies, they will put these towards the cost of the funeral. They do all the filing at no extra cost. We had the clothes for my Mom. We also had a picture so they knew what she looked like before everything at the hospital. They will order any death certificates you need. They only charge you what they are charged and it is one less thing you have to do. They will also handle ordering any future ones for you if you prefer. If you give them your pastor's information they will contact them and if your pastor does charge for their services they will add that in...we contacted our's since we had 2 involved and they don't require payment. I needed more thank you cards than they gave in the set we ordered, you may want to see if you can order more. Our funeral home just gave me another box. We didn't do an obituary, my Mom's choice, and it was a private service.
As for cemeteries, most require full payment of the opening and closing of the grave before they will start anything. We were fortunate that my Mom had already done this and had already picked out the stone, she just hadn't ordered it.
Social Security has probably already been notified, but you can call your local one to make sure. They will send your sister a letter telling her to call them regarding the one time Survivor payment of $255 and to discuss any other benefits for the future.
If your sister had accounts with her DH they may be locked by now until everything is taken care of. I know at my sister's bank they go through the death notices every day and lock the accounts until any estates are settled.
If their is anything that is not standard your sister wants, don't hesitate to ask. My Dad requested to stay till the last dirt was in and they allowed us to even though machinery was used. They also wasn't going to let us back because of ice on the roads, but they knew how my family was and still let us back.
One part that I really liked at the funeral was that before they closed the casket we were allowed to 'tuck' my Mom in. We put the fabric that was outside the casket into it and covered Mom with an additional piece on top of the casket. They actually offered this at the last minute and I am so glad they did. Not that it is for everyone.