I am very sorry. I lost my dad too and had to do all this after he died. I missed him by 5 hours and my mom and I stayed up all night writing the obituary because she had not done that.
If you don't know where the will is and if you don't know about his life insurance policies, ask him in a lucid moment. You need to find those policies, and the will is incredibly important. Any other important papers, lock boxes or brokerage accounts, too. You need his social security card, retirement documentation, drivers license, etc.
If your mom and dad have a joint bank account, she should go take money out of it now, before he dies.
You'll need the funeral director to order death certificates from the county right after he passes. After you get them, you can get Letters of Testimentary from the court so that you can start executing the will. If his car is only in his name, you have to bring evidence of value (Blue book value) to determine if the car will go into probate or if it can just be put in your mom's name. Make sure you have the will, the title of the car, etc with you when you go to the court. Make sure to take him off the car insurance policy.
Make sure that you send COPIES (not originals - they'll just send them back) to all three credit reporting agencies and a letter to each one detailing that he is deceased and they need to flag his social security number and his accounts so that no one can try to take his identity to get credit. You wouldn't believe how much that happens.
Don't put your dad's full birth date in his obituary. If you put too much personal info in the obit, it helps fraudsters to pretend that they are him.
Right after the funeral date is set, get on the phone and call people for your mother. She probably has an address book with phone numbers. The faster you can call people, the more chance that they can drop what they're doing and come to the funeral to help support you and your family. This will really help your mom. It will be hard for her to make these calls.
Try to find someone to stay IN the house during the viewing and the funeral. Sometimes people look at the obits to see what time a funeral is and they break into the house during the viewing or funeral. That's really terrible.
The hospital has social workers who you could talk to about helping you emotionally prepare in these last few days. I sang to my dad over the phone to help us both. I couldn't do any of the business stuff until I arrived. Thankfully we had a lot of ducks in a row for a long time before that since my folks are elderly.
I found that staying busy with all the paperwork helped me. I stayed on the phone with my mother for an hour talking to the credit card people to get my dad off the account and my mom on the card herself. That can't be done until after the funeral. I took her to the bank to open up the new bank account (you MUST do that because it's illegal to use a bank account if someone is deceased.) You need the Letters of Testimentary to do that, which means the funeral home has to get the death certificates to you. (Double check that death certificate before you use it. The first ones we got had Dad's social security number wrong and we had to order new ones.)
Ask the funeral director if they help file for the life insurance benefits. Our's spent 4 hours going through my daddy's policies, calling on the companies on the phone, helping me and my mom navigate the entire process. They made all the copies and mailed them certified to the insurance companies for us. It was a big job and we were so glad to have a CLEAR set of eyes overseeing it so that it wasn't just us. I'm sure that your funeral director's people will do it too. Just don't sign over the life insurance policies directly to them. An unscupulous outfit will keep the difference in what is owed and what the policy is for. I'm not saying yours will, but still, better safe than sorry.
Make sure that you double check the spelling of EVERYTHING that goes on the grave. Since you're Jewish, you may not have a headstone for a year, but make sure that the spelling is right. Sometimes when you're grieving, you look at what is written and don't see mistakes. There's a plaque at my dad's cemetary that the name is spelled wrong and the mistake was made because the family didn't re-check.
I know that none of this sounds comforting. I'm sorry. I just want you to know the things that were important for us, both before and right after. I only had a week with my mom after my dad died. I had to make the most of it.
My best to you through this oh so hard time.