Yes, there are private IP addresses in IPv6. I'm shocked that no one has objected to answers that claim otherwise (and I don't have enough reputation to comment).
They are called Unique Local Addresses and as of 2018, they are basically limited to fd00::/8.
They cannot be used with NAT the way IPv4 private addresses are used, however.
Yes, NAT does actually exist in IPv6, but not in the form of the "masquerade NAT" used in IPv4 (it serves a different purpose). It is called NPTv6 and it's used for prefix translation.