Continental Gator Skin, Schwalbe Marathon are good puncture resistant tyres.

Punctures are a normal thing. They will happen with puncture resistant tyres as well. When it's dark, raining and you're away from home. Always carry a spare tube and a patch kit.

The wider the tyre, the lower the pressure, the less chance of sharp objects puncturing it. Best way to avoid punctures is watching where you're riding as much as possible and having lower pressure, wider tyres. Anything below 6 bars is pretty much resistant to sharp rocks and even glass (in most cases). Anything below 4 bars (that would call for 37 mm tyres, so not possible for road bikes) is pretty much bomb proof, even when using regular, not puncture resistant tyres.

Puncture resistant tyres ride more harshly. I'd use them if I rode regularly through roads full of thorns or glass, so it's not possible to avoid them. Otherwise, not worth the comfort penalty, IMO.