Medulla issues?

My father (73) suffered a double stroke and has been bedridden at JLL for 110 days now.
The CT scan shows a left perivascular ischemic lesion. Curiously, his motor skills seem intact—both arms and legs move—though there is a slight delay on the left side.
He’s on a ventilator, though not constantly. The doctors say the damage is in the medulla, which explains why his breathing stops—usually while he's sleeping—and why he can't swallow, necessitating the stoma.
He’s already fought through aspiration pneumonia, an anaerobic staph infection, and frequent seizures.
Out of those 110 days, he’s only had a speaking cannula for three. He remains conscious and communicative, but it's clear he's disoriented and essentially constructing his own reality.
I don't have a medical background, so some clarity on these neurological issues would be invaluable. For instance, would a CT scan immediately pinpoint lesions in the medulla area? Why does his breathing stop in these predictable cycles every seven to ten days rather than being random? And why would a lesion on the left side of the brain cause a delay on the left side of his body?

Thanks.

Perivascular or periventricular? Periodic respiratory failure can stem from various causes, and it isn't always strictly neurological. Which department is JLL's father in? Anesthesiology or neurology?

Excuse me, periventricular. Regarding anesthesia—does the ICU at Mayo Clinic not have ventilators?

That single sentence says it all; that final point alone proves it isn't intense, but let’s move past that for now. What exactly do you mean when you say one side is lagging? Regarding the respiratory arrest—is it a sudden stop, or does it just become increasingly difficult until they are once again dependent on a ventilator?

They call themselves the intensive care unit, while the anesthesiologists are... well, that's all I know.
Respiratory arrest happens out of nowhere—"it just flushes and stops despite clear airways," they claim. Or more often during sleep, it just ceases. One moment they're talking, the next their blood pressure spikes insanely and they stop breathing. It's as if the respiratory center was simply suppressed. Then comes several days on a ventilator—on, then off, then back on again. Excessive secretions.
No paresis—just faster and spontaneous

The right side of the body responds more spontaneously.

Is the left side lagging behind? As in, is it developing more slowly? And how are you feeding it?

Yes, she’s slower. She has a gastrostomy tube—can’t swallow at all.
A few days before she stops breathing, she gets restless. Doesn't sleep.