I'm in Iceland right now with my kids. As of now until July 1st, everyone who arrives in Iceland gets tested on arrival. Kids just do whatever the adults in their party do. If you are vaccinated, you just do the one test and quarantine until results come back, ours took about 5 hours. You also register with covid.is 72 hours before your trip to get a barcode for your test. Everyone in the group needs a barcode. Similarly, you register to schedule a test returning home. They are currently planning to stop testing vaccinated visitors on July 1st, including the kids coming with them not getting tested, but that could easily change.
I know of one person who got a false positive when they arrived, they just got a second test that day and it was negative.
You are extremely unlikely to get a positive test returning to the US if things stay the way they are, there are very few active cases of covid in Iceland.
I was definitely very stressed about getting here with the kids, very worried about the tests and what we'd do if we had to quarantine the whole trip, so I don't think you're over thinking it. Or, at least, if you are then you are overthinking just like I did!