At my daughter’s high school, 6 classes is standard. Students can take 7 if they want, but most students take 6 and that is considered a “full schedule.” I honestly don’t think it matters if you take 6 or 7 classes, as long as they are fairly rigorous ones and you do well in them.
Colleges generally don’t verify ECs. Theoretically they can (and you would need to provide some sort of documentation), but it is pretty rare. They usually take your word for it.