A few things:

1. A stroke does not equal death. A stroke is either a blockage of asupplying artery to the brain (ischaemic stroke) or a rupture of –most often– an aneurysm of one of the intracranial arteries (haemorrhagic stroke). If left untreated, part of the brain dies off, but that is entirely survivable. Old people can have multiple holes in their brains from minor strokes and still walk around. If a large area is affected the patient may become hemiplegic and will need help with many daily tasks and activities, but a stroke generally doesn’t cause death. There are a few exceptions though (see 2.).
2. A person generally doesn’t just stop breathing with a functioning heart and brain. A rare cause could be a basilar artery occlusion or haemorrhagic stroke affecting the midbrain’s respiratory centre.
3. We don’t really have a lot of experience with artificial hearts that provide pulsatile flow through pneumatic pumps, the few that have been implanted were mainly for research purposes. What we do have experience with is LVADs, or left ventricular assist devices. These are rotating centrifugal pumps that are implanted inside the heart and aorta and provide laminar, continuous flow. These pumps are so strong they can keep the brain oxygenated even if the heart itself stops pumping (e.g during ventricular fibrillation).
4. As for your question: death would be brain death or multi-organ failure. The blood would still flow and provide oxygen, but the end organs (including the brain) would not have enough functioning tissue to sustain life. The LVAD would keep pumping until it runs out of battery or is shut off.

Most countries use brain death as the deciding factor.
Meaning you body can be held alive artificially, but if the brain isn’t working, you are dead, and there is no chance of coming back.
Of course if the heart hasn’t been beating for 5 minutes and you are not kept artificially alive, you are probably considered dead too.

I would assume that it might keep going until it dies if it’s battery powered. As for non-battery powered, like the replacement valve in me, probably gonna stop when the blood stops flowing. So gonna assume it’s the same for non-battery powered mechanical hearts as well.

I would imagine that it would. The alternative is that it would not, which would mean that it would need some sensor to determine that you were no longer alive, at which point it would stop working. There’s no real point for it to have a feature to stop working. It’s probably better for it to be seen as ‘creepy’ by continuing to work inside of a dead body than for it to constantly be checking to see if you’re alive, since the surgery to change the battery is probably pretty invasive, so they want to do it only every so many years, to maximise its efficiency. Also, false positives (negatives?) would be problematic. You’d just want it to keep working regardless.

EMTs probably have a way to shut it down, though. It’s probably impossible or really difficult to do it accidentally, but I’m sure there’s a way. Good question for an EMT if you happen to meet one. They aren’t squeamish. It’s just a matter if they feel like talking to you or not, I suppose.

Depends on how it’s meant to work. If it amplifies the actual heart’s movement, it might not do much. They usually have a power supply, which will make it keep doing what it does.

So, if it’s a 100% artificial heart, it will do what it does. If it’s there to force the heart to beat without the heart’s natural movements, it will keep force it to beat the same as before, same as if you pass electricity through any muscle.

I happen to have some experience with this, as I used to teach anatomy at university. Many of the corpses that we used, contained metal knees, and other improvements, you can, for instance, clearly see where open heart surgery was used to improve the heart. Now, all these corpses have been soaked in alcohol and formaldehyde to conserve them, so they are obviously dead. A few times however, the olde pumpe may start beating again, if it has artificial components. We tutors laughed our toupees off, as students would get the shock of their lives, when the corpses would sit up, or even chase them across the room!

A cold, monotone voice can be heard: “this person has expired. Please contact the authorities.”

This isn’t what you asked exactly, but I know they can externally disable a pacemaker once the patient is determined to be deceased so it will stop trying to shock the heart back. Otherwise it will continue to try to save the person.

Your body is alive but your brain is dead and that makes Jesus sad.

https://youtu.be/DsY_kocbWaM

Time Stamp 1:33