Hello, I have a question about flash of genius.

FLASH OF GENIUS
7th-level artificer feature
You've gained the ability to come up with solutions under pressure. When you or another creature you can see within 30 feet of you makes an ability check or a saving throw, you can use your reaction to add your Intelligence modifier to the roll. You can use this feature a number of times equal to your Intelligence modifier (minimum of once).
You regain all expended uses when you finish a long rest.

For an ability check, is it using skills you're proficient in, using tools you're proficient with, is it right ?Is it also with initiative because it's a Dexterity check ?

The feature doesn't set a limit on the type of check, so I would assume it can be any check, including the check made for initiative, and without limitation to skills/tools you are proficient in

Yes - Initiative is a Dexterity Ability Check. As are Dexterity (Stealth) Ability Checks - or Charisma (Intimidation) - or even Strength (Intimidation) - or tools and instruments. Proficiency is irrelevant. An ability check is an ability check whether you add your proficiency or not.

I highly suggest you refresh yourself on the rules:
https://www.dndbeyond.com/sources/basic-rules/using-ability-scores#AbilityChecks

Every time you roll a d20 and add an ability score modifier, except saves and attacks, is an ability check. This includes initiative, all skills, certain tool checks, and just plain ability checks.