Here are two pictures that should explain everything.
To your question, "Also, wouldn't there typically be more notes in between the bass clef and treble clef? Why are they missing in this case?" I should explain that in typesetting sheet music, the amount of space between the treble and bass clef of the piano is variable, but the number of notes between them are not. More or less space can be put between the staves to allow for slurs or ties or other musical symbols or text, and to allow for the peculiarities of cross-staff notes and cross-staff beaming. Regardless of the visual appearance, technically there is only one ledger line between the staves, and on that line goes only the pitch C (or C# or Cb).
It should also be noted that the two staves are traditionally set up to separate the part played by the left hand from the part played by the right hand; the lower staff is for the left hand, the upper for the right. When the music requires both hands to be in the bass register, or both hands to be in the treble register, the two staves can use different clef signs (for instance two staves with treble clefs or two staves with bass clefs) or the notation "8va" or "8vb" can be put over certain notes to indicate that they are to be played an octave higher or lower than they appear on the staff.
When you see notes like the example below on the left, this is to indicate that the note below the treble staff is to be played with the right hand. The example on the right notates exactly the same thing, with exactly the same pitches, but using cross-staff beaming.
Here is an example from a Bach harpsichord concerto.
The two musical lines can cross from one staff to the other, but because of the cross-staff beaming and stem direction, and the lack of rests in the second measure in the bass staff, it's obvious that there are two lines played by two hands.