Yep, that's just trying to sell you an optional private insurance. It's nothing to do with the NHS.
It's very sensible of you to check though. If I was moving to another country, I'd want to check and double-check everything!
Yes, you're entitled to live and work in the UK and you shouldn't have any problems. But very occasionally EU citizens do have to provide extra proof to get things like UK welfare benefits ( for example, financial help if they're not working). I work with people claiming benefits and I remember two people who did have a few problems. One was from Eastern Europe and the other was from Portugal. It's not common, but it's best to be prepared. I really don't think you'll have any problems at all, but I think it's nice to know about potential problems even if they're rare. So bring all your documents and that should make it much easier. Your English is brilliant so that'll help a lot too.
You sound like you're very organised. I hope you settle happily in the UK and that everything goes well for,your move