The 23-year-old Mina, who was impressive in defence at this year's World Cup in Russia where he also scored three goals, will rejuvenate an ageing Everton back line while offering a threat from set-pieces at the other end of the pitch.
The fee could rise by 1.5 million euros and Barcelona have inserted a buy-back clause in the player's contract.
Everton's season-long loan deal for Portugal's Gomes will see them pay 2.25 million euros and the player's wages. The 25-year-old has played 29 times for his country.
Bernard, also 25, has signed a four-year deal with the Merseyside club after his contract at Shakhtar ran out in June.
Capped 14 times by his country, Bernard can play on either wing or as an attacking midfielder.
The Brazilian said new Everton boss Marco Silva convinced him to join the Toffees.
"The manager made me feel confident about playing here. He is a very good coach and he has achieved really good things at other clubs," he said in a statement.
"Everton is a club with a great structure and a lot of tradition, historically the fourth most successful club in England. When I found out about this opportunity and what the manager wanted out of me I was happy."
(Reporting by Hardik Vyas in Bengaluru; Editing by Ken Ferris)
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