@Earlydarkdays oh much love to you and condolences on the death of your father.
Your user name is spot on because it is a horrible horrible horrible time. I've been through this a while back and I would say
-the death of your parent is one of the toughest emotional experiences you will go through in your life. People often say grief is the price paid for love and there is a lot in that. Do try to hang on to that sense - the pain is a reflection of your love. Even if it is expected, its a shock. if it was unexpected, the shock will be multiplied a gazzilion-fold. So first thing is don't expect too much of yourself (or your mother) for at least 12 months and possibly longer.
-Do NOT make any big decisions (her selling the house/moving etc) for at least 12 months because you aren't functioning normally.
-You are in the worst of it. It's all raw and the pain is so searing and you are in shock. This time will seem like a blur in a year or so because of the shock but you will remember the kindness and the friends who stuck by you. Grief makes many people disappear.
-You and your mother are in the same boat - having lost the same person you both loved - so you can help each other in your grief. BUT your grief will be different because you were a daughter and she was a spouse AND because everyone grieves differently. There is no hierachy or judgment in grief. It all hurts, its all different.
-If she wants to be with you but you need space, my advice would be to manage that as best you can to give her what she wants at the same time as getting what you need. If her grief means she wants to be with you all the time, honour that and do that as much as you can because it will comfort her short term and long term she will remember it and you won't judge yourself adversely/feel guilty later (not that you would have a reason to but we all have capacity to feel irrational guilt). HOWEVER, your grief is important too so if you need some space, manage it as best and as gently as you can. So for example rather than saying to her 'go back to your own house for the afternoon', I need time to sort out the funeral' find an alternative - can you contact a friend of hers or another relative and get them to sit with her for a bit or do something of low engagement and intensity - like go for a walk with a dog.
-A very useful piece of grieving advice I was given for spouses/people who live together is to get the spouse to move into the deceased's usual spaces in the house - by which I mean if DF had a favorite chair, DM should now sit in his chair and if he had a particular side of the bed, she should sleep on his side of the bed. This has two massive benefits, it avoids the discomfort and upset of sitting there just looking at an empty chair or waking up and looking the direction they've always looked to the empty side of the bed. Secondly, it can allow them to feel closer to the deaceased.
-Don't forget yourself. A book I'd highly recommend is
Coping with Grief 5th Edition (Paperback)
Dianne McKissock (author), Mal McKissock (author)
https://www.waterstones.com/book/coping-with-grief-5th-edition/dianne-mckissock/mal-mckissock/9780733339578