Class exists, it's just about a different set of behaviours.
Yes but it isn't set in stone the way it seems to be in the UK. So you don't have the strange situation of two university professors, living in a 4 bedroom house in the right part of town, with 2 children at grammar - or maybe independent - schools saying "we are working class" because their grandparents or parents worked in a factory. No one in the US would say this but you hear it all the time on MN. Also the cliches about how you can tell what class people are by how much they clean their houses. It is very strange if you didn't grow up there.
On the baby name thing, there is a strong tradition in the US (particularly in the South but not just the south) of giving a child his mother's maiden name as a first name - hence so many names that are surnames. I kind of like them and like that it gives a link to the mother's family and heritage.
Most of my children's friends (they are older teens) in the US are called things like Alex, Max, Emma, Julia, Sophie, Isabel, Emily, Alex, Alex, Alex (did I mention Alex - one year I think the entire hockey team was called alexandra and its derivatives), Joseph, John, and a few names from people's heritage - like Nilufahr (my favourite name), Rajan, etc. My dd is friends with a Navaeh - only one we have met - she is a great kid.