Well this is a lot more fun than studying for my upcoming exam.
Depending on how much caffeine you drink, caffeine will decrease your heart rate (low dose), increase it (high dose), or cause the heart to beat irregularly by directly bypassing the heart's internal pacemaker and causing heart muscle cells to contract out of sync (extremely high dose). Citation: the end of paragraph 2 on page 2. In people who don't have any underlying abnormalities with their heart, caffeine is probably fine, but it's not uncommon to have an abnormality that you don't know about, and if you do, significant caffeine intake could provoke an issue.

Is it not common knowledge that acids (including carbonic acid) can cause the demineralization of enamel? It's the same reason why we tell patients with GERD to take their PPI/antacids: if you don't, your teeth aren't going to look great (and you're at an increased risk for esophageal cancer). Anyways, if you want research to support that, a quick search reveals in vitro evidence showing that exposure to Coke, Diet Coke, Coke Life, or some Zevia Cola causes extracted teeth to erode more than exposure to 0.9% NaCl (incidentally, a similar study done with extracted cw teeth showed that Sprite, apple juice and OJ are 5x more erosive than Coke Light, so which non-still water you choose to drink matters). Interestingly, another study on extracted teeth (you'd need to look at the full text for this one) showed that sparkling water is worse than still water but nowhere near as bad as soft drinks, likely because there are other acids in soft drinks, some of which are probably stronger than carbonic acid. So Adam's dentist is right: soft drinks are bad for your teeth, and carbonated water is better but not as good as still water, but everyone has their own calculation on whether they think it's worth it or not.
When they say it's sugar-free, they mean it has nothing that your body will metabolize as sugar (as far as they know). It still has chemicals that our bodies might not take in...but bacteria may still like. Drinks with "natural sugars" have the same potential for acid protection and can still generate cavities.
Lol, not quite, but the big risk factor (aside from the sugar stuck on your teeth giving something to eat) is the fact that some drinks will bring your mouth's pH below 5.5, which is the point at which enamels break down. It'll slowly go back up as more and more saliva is produced, and you can probably accelerate that by drinking tap water