Regarding the amounts of PTO, it depends on the company. Some companies don't offer it, so if you get sick and can't go to work then you can stay home and get better but you don't get paid. Those jobs suck and from what I've seen those jobs have high turnover due to people going to other companies for better benefits, like getting PTO. Most companies offer it, but in limited amounts. I personally get 5 days a year of paid time off, not including vacation time/LOA/Floating holidays, but I've utilized 11 days this year, I just didn't get paid for 6 of them. I think there are some companies that offer unlimited PTO, but I would guess those are only for experts in their fields and stuff.

PTO isn't requested, it's notified, as you described. If I get sick, I don't ask my employer if I can take the day off, I tell them I'm not feeling well and I'll be staying home and utilizing PTO. That's how it's worked for me at multiple different jobs.