In ADFS do we still need to type the domain for authentication?

you can either use SamAccountName in format domain\username or you can use userprincipalname (user@domain.com)

Both are acceptable to ADFS

If you are looking from O365 stand point, its better to user userprincipalname and probably user principalname would be equal to user primary smtp address

But let say I have an application that we use with ADFS? How can I connect to the application?

1- I need to go on ADFS URL first and authenticate myself
2- After that I can start my application.

Does it work like that?

You can customize the page to enter only username, refer https://blogs.msdn.microsoft.com/sakamati/2015/06/22/customizing-adfs-3-0-sign-in-page/

Thanks