How can the key of C have an A7 and an Fm in?

The musical key is a starting point, a foundation if you like. Beyond that, you can be as creative as you like to create the harmonies or tensions you are looking for in your tune. A good example of venturing into related keys and mixing majors and minors of the same chord is in the Dylan song made famous by Adele 'Make You Feel My Love'. Depending how you choose to play it, that song in the key of C includes G, B flat, F, Fm, D7 and E7. I would use a few extras in the transitions  but, even at its simplest, this song demonstrates how musical keys should not be considered a straitjacket.

In simple terms, a piece of music in the key of C “can” include any chords. It's NOT uncommon for music in C to include chords that use notes that are not in the the C major scale.

Music in any  key can have accidentals, which indicate notes that are not part of the scale for the key signature.  Sometimes there will be a "chromatic" scale in music, such as Flight of the Bumblebee, in which every note is played, sometimes for part of an octave, sometimes for one or two octaves.  Listen to Emily Bear playing Bumblebee Boogie.  I find it fascinating.

What you learn in music theory applies to 18th/19th century music, not to modern-day pop music.  In pop music, things like using A7 in the key of C are very common.  As another answered said, in pop music, intervals are everything.  C to A major, C to A minor--both very common.

Btw, A7 does not have any flats.  Has a sharp.

The A7 chord is the dominant of D - in this case, d minor, since you are in the key of C, which had an F natural, not a F# - unless you ALTERED another chord, like using the F# in a D7 chord to get to you to G - the V chord (that process is is called using a double dominant.  V of V.)  I would have to see/hear the song to which you refer to give you a solid response about the fm - it could be only a momentary change for *color*, but it *could* - and you do nor give me what happens after that fm chord - a part of the most obvious progression - fm - Bb7 - Eb.  That is a modulation (longer-term move to another key) or temporary invasion (does not stay there long - gets back to home key fairly soon) to the lowered third - a very, very common, almost obvious, usage in pop music, and some MoTown and musical-show music.  I was just trapped in my chiropractor's waiting room this morning, where he had some kid of HIDEOUS music feed playing - and song after song was some country-pop crap, that used ONLY the tritest, most nauseating and predictable chord progressions ( and cheesy melodies, and lyrics, too . . at least the voices were good.)  I  made him turn it off in the treatment room.  In the waiting room, had earplugs in my bag - but had not taken my earbuds, to drown out such hideous junk.  So the two examples you gave here, might be part of something creative, and used tastefully - or could have been part of something that would cause nausea in any decent musician.

Yeah - I am a theory teacher, for over 30 years - multiple degrees in that area, from fine conservatories.  Makes the GREAT stuff more of a thrill - and the rotten stuff, just more of a trip to the rings of the Inferno. God save us from crappy music . . .

A is the relative minor of C.  C is the perfect fifth from F.  Intervals are everything.