The violence -- throwing the cat (a very big red flag: he's willing to hurt something that's living), jumping in the car like that, "pushing and shoving" -- frankly that would be enough, right there, to make me have my mom tell him today he either gets to a professional, today, or she sleeps at your house or a hotel tonight. And every night after that until he has:
--Seen his doctor and had a full workup of all his medications. You said he had had a med change; that could be creating the anger etc. And why is he taking Ambien? Why just one, that one time? If it's not a med he uses regularly why did he pop it that one time? This could all be related to side effects.
--Seen a therapist. Not a grief counselor -- it's great that your mom is going but it does not sound appropriate for him. He needs first the medication review and then a referral to a very good therapist.
--Gotten checked out thoroughly by a doctor who is a specialist in early onset dementia. Yes, I'm talking Alzheimer's here. I would probably not use that word with him yet as it might set him off and make him refuse to talk to ANY doctor. But the things you describe here -- irrational outbursts, anger (even at the person you love most), violence -- can all be signs of the onset of dementias. Dementia alters the personality. It may not just be resentment and anger at your mom's caring for her mother; it may be that coupled with the early stages of dementia.
Please don't think, "At 60 he's too young for it to be Alzheimer's or another dementia." Not so. My friend's mom was diagnosed with early-onset Alzheimer's at 60. And please don't think he's incapable of real violence. My mom's neighbor was the gentlest man in the world, but Alzheimer's made him have very violent thoughts.
This may be as simple as medications gone wrong, or it may be as profoundly life-changing, for all of you, as his having dementia. But you cannot know unless you get professionals involved, and quickly. Not knowing your relationship with him, I can't say "You have to take over, make doctor's appointments, and tell him you will be driving him and sitting in taking notes on every appointment," etc. But that may be what it takes if your mom is not in a position to handle it, or if he would become angry at her doing so. It sounds like he cannot be depended on to see the problems in his own behavior or to take action to stop it.
Meanwhile, I'd convince your mom that she may be in physical danger from him. I find the things you describe -- especially if they are sudden and not part of years of violent outbursts -- very worrying.