After everything it indeed sometimes feels like it comes down to this. They want to see you physically sitting at your desk for at least 8 full hours, that's the only 'measurement' in productivity that most companies seem to care about.
Sick days do get logged, and if I really need more time off I can start burning vacation days. Given the choice, I'd rather spend a shitty sleep deprived unproductive day at work.
Eventually goes up to 10 sick days, but I don't remember when the threshold is. Frankly it's a strange thing to base on seniority, as if newer employees somehow get sick less?
Email/call in sick, stay at home. There is an implicit "stay a little later when you're feeling better and/or make up lost time on the weekend". As long as nothing falls behind it isn't an issue. That's the informal policy. If you were ahead or don't fall behind - the extra hours don't need to be picked up. The only important thing is that things remain on schedule.
Hungover, sick, just don't feel like coming in that day, doesn't really matter.