iPhone 14

1. This is is a user to user forum, you are not addressing Apple or apple support directly here. We users, do not fix screens at all.
2. Yes, Apple is the best option to get it fixed as it guarantees you get a genuine screen, and avoid any potential issues. click here ➜ About genuine iPhone displays - Apple Support
3. Yes, unless there's any other damage to the phone, screen replacement is usually done while you wait.
4. Click here ➜ iPhone Screen Repair & Replacement - Apple Support

Start here.

iPhone Repair & Service - Apple Support

AbNiKe wrote:

Also, do you fix screens in one day?

I have never fixed a screen in my life. I, like most others here, am a volunteer answering technical Apple questions.