I will say I'm not too surprised at the updates.
It's a pretty common pattern - you've gotten fed up, and are trying to enforce boundaries on behaviour you previously put up with. The person behaving badly 1) takes this as a challenge - they figure they need to push harder to get what they want and 2) are testing the boundaries to see if you mean it.
Dealing with this can be surprisingly like dealing with a toddler. You need to stand firm, or they'll quickly figure out that you don't really mean what they say. And you need clear, immediate consequences for their bad behaviour, so they learn that if they do X, Y will happen, and then don't want Y to happen.
If your Dad keeps pushing, I do think that blowing up and getting mad might help. Get exasperated, raise your voice, tell him that you're sick and tired of being pulled into the middle of their marriage and you're not going to put up with it. You're not going to listen to his complaints, you're not going to watch him sulk, you're not going to hear his armchair diagnosis of your mother's mental state, you're not going to offer advice on legal or financial issues, and if he doesn't respect that, he won't be talking to you at all. Then stomp out of the room and ignore phone calls for a while.
As an aside - one thing you might find useful for your own sake, if it's feasible, is having a few sessions with someone with a speciality in family counselling. Tell them you're not interested in fixing your parents' problems, but you want to learn how to avoid getting pulled into the middle, and how to deal with them most effectively when they're being difficult. A specialist will have seen dozens of situations like this, and know what can work and what doesn't, and what the dynamics are.