It is a little difficult to answer your question without knowing your understanding of theory. C'est la vie. I will do my best.
The first progression is fairly common: F, A♭ dim7, G min7, C7. F is the tonic (I) chord, the G min7 is the II chord, and the C7 is the dominant (V) chord. The function of a dominant chord is to lead you back to tonic. In jazz, the II chord is a chord that leads you to the dominant chord to provide harmonic interest preceding the V chord. In this progression, there is a Ab dim7 before the II. This occurred frequently in earlier jazz to add chromaticism to the progression. Non technically, it sounds bluesy.
The second progression is a transposition of the first.
In regard to seventh chords, jazz almost always uses seventh chords or other added note chords. Most chords with a '7' are constructed with a minor seventh above the root of the chord. The only exception is a FULLY diminished seventh chord, which has a diminished seventh (enharmonic with a major sixth) above the root. If there is a major seventh above the root, the chord symbol will indicate that (G maj7 or [less frequently with minor chords] A min maj7).
In the third progression, C, C♯ dim7, D min7, G7, the C♯ dim7 chord, being a half step below the D min, is leading you to the D min7 chord, and is a substitute for a dominant chord of the D min (very common in baroque music as well).
You might get a clearer answer by (if you have not done these things already) studying a general music theory book, studying jazz theory specifically, and playing many jazz tunes. Aebersold play alongs are great sources for jazz tunes. Bon chance!