I would give warning to my current employer, because I've been here for a long time, they've treated me alright, and I have a lot of knowledge that's important to the business. But I'd see it as a courtesy, not an obligation.
The professional relationships you establish today can be very important in the future. Your coworkers can give you references, help you get hired, or even do the hiring themselves one day. Therefore, it is in your best interest to not be an asshole and leave them hanging by quitting without warning.
There is always ways to make things work when relationships are good.
Most people have little reasons to switch jobs if they are happy.
Lately, I've trended towards not trying to work with people at future jobs I've already worked with as I don't like the pressure of having to live up to whatever expectations they or others have. I'd rather someone have reasonable expectations that I can actually beat and vice versa.
With or without two weeks of pay? I don't think I'd fire someone without two weeks of pay in lieu of notice, although I suppose if the behaviour were terrible enough it's possible.
After all, you will have to be able to cover someone if they are sick (typically short time, no notice) - while presumably replacing an employee fully will take longer than two weeks anyway?
Agreed. 2 weeks feels like a number so small it effectively rounds to 0. I am actually unsure of the minimum notice period here (Australia) but I have typically given 1-3 months notice unless there was a very good reason why I needed to leave sooner.