You're way over thinking a fundamental and simple jazz concept.
Please break the association in your head that "dominant 7th chord = mixolydian mode." That's not true, and no jazz musician that I know thinks about music that way. It might be true in certain other styles of music, but definitely not in jazz.
The symbol "A7" is not shorthand for a "scale" or "mode." Rather, the symbol "A7" is shorthand for a "chord." In this case, it is the chord A7, which is spelled A, C#, E, G.
Jazz composers will freely use lots and lots of dominant 7th chords. There is no requirement that these dominant 7th chords fit any "parent" key, scale, or mode. Rather, these chords are used to create "tension and release" which keeps the harmonic progression moving along.
I bet you that the next chord after the A7 is some type of E chord. In this case, A7 is acting as the "secondary dominant" of E. You could say "five of E" and most jazz musicians would know exactly what you are talking about. Here is an article describing the basics of secondary dominant theory:
Secondary dominant - Wikipedia
I realize I haven't answered your specific question, "should I play F or F# over the A7 chord?" My answer is that, in the "jazz" style, it would be perfectly appropriate to play F, it would be perfectly appropriate to play F#, and it would also be perfectly appropriate not to play either F or F# at all. The four important thoughts I try to keep in my head:
- What is the harmonic chord progression of the song? What are the chords before and after? Am I choosing notes that connect smoothly from chord to chord? Am I providing an appropriate amount of tension and release?
- What is the melody? Am I choosing notes that support the melody?
- What are the soloists doing? Am I choosing notes that support their solos?
- What ideas can I get from classic recordings? How would Ray Brown play a progression like Dmin7, G7, Cmaj7, A7? What about Ron Carter, Jimmy Garrison, or Jimmy Blanton? What notes would they choose for this chord progression? The best way to become a jazz musician is to study jazz songs by master jazz musicians. Don't get lost in your own head; put on some famous recordings and jam along with what you hear!