To answer your second question first: If you were to play an alto recorder reading the music as if you were playing a soprano recorder, and you were playing with someone else playing an alto recorder using the f fingerings, then, yes, you would be the one who sounded way off.
As far as an alto recorder in c is concerned: If you have a recorder that goes all the way down to the c below the bottom of an alto recorder, then what you have is a tenor recorder (which is pitched in c).
I suggest that you immerse yourself in learning the f fingerings for a while. Learn them for what they are. Then, when you have alto as actual alto down, you can pick up your soprano again and you’ll know both the soprano/tenor and alto fingerings.