The form on HIBP uses the same JS client hashing, you can check the HTTP requests yourself in dev tools.
Yes, the whole dataset is available. The first paragraph mentions the release of the v2 dataset and you can read the full blog post here: https://www.troyhunt.com/ive-just-launched-pwned-passwords-v...
You can get the 8.8gb file directly here: https://haveibeenpwned.com/Passwords