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Gelhardt has signed a four-year deal with an option of a further 12 months, with Gourna-Douath joining the Tigers on a five-year contract.
The pair become Hull's 10th and 11th signings since securing promotion back to the Premier League last season.
Gelhardt, 24, returns to the club after two successful loan spells over 18 months which saw him score 14 goals in the Championship last season and the tie-clinching second goal in the play-off second leg against Millwall.
He started his career at Wigan, before joining Leeds where he scored three goals in 57 appearances around loan spells at Sunderland and Hull.
"I'm over the moon. It's been a great 18 months playing here - I've loved every minute of it. Since the season ended, I knew exactly where I wanted to be. Thankfully, it's happened," said Gelhardt.
"As soon as I joined the club, I connected with the fans, the staff, all the players. I'm happy to be here on a permanent now, rather than a loan."
Gourna-Douath started his career at Saint-Etienne, where he made 65 appearances for the French side before joining Salzburg in 2022.
He made 95 appearances for Salzburg over four seasons and spent the second half of the last campaign on loan at Le Havre, where he played 15 games in Ligue 1.
"I'm really happy because it was such a long journey from the first phone call until I signed. I'm so excited to get started," said Gourna-Douath.
"I know Hull City is a big family club. Playing in front of our fans will be fun. It will be amazing for me."
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