Typical Roles of the Left Hand
- Bass line
As you mentioned, the left hand almost always plays the bass line. Some expressive pieces, especially from the romantic era, have a very subtle, or practically non-existent "bass line", so "left hand" and "bass line" are not perfectly synonymous.
Rhythm is another thing the left hand often plays (no pun intended) a large part in. The downbeat and pulse are often (though not always) given by the left hand while the right hand is more likely to play runs, ornamentations and syncopation. (Once again, more advanced and expressive pieces often don't adhere to this at all).
- Additional voicing
In addition to the bass line, the left hand can add more voicings or harmonies, much like the right hand often plays more notes than just the melody. These middle voicings (what you might call alto and tenor) are crucial for certain voice leadings.
Additional Benefits of the Left Hand
It is much easier to play the hands independently (with distinct rhythms) from one another than it is to play fingers on the same hand with independent rhythms.
- Greater range
A single hand cannot span much more than an octave. With two hands, any two notes can be played simultaneously.
- Open chord voicing
Related to the above, the left hand allows chords to be spread out over more than one octave (I.e. C-G-E-B instead of C-E-G-B). This is often more desirable and allows for greater diversity.
As for your second question
When in doubt, you can always play the root of the chord. To spice it up a little you can try playing a different chord tone, creating an inversion without altering the harmonic structure. From there you can try inserting passing tones in between chord transitions. If you're feeling adventurous, you can even try playing non-chord tones, giving you a slash chord (I.e. Dm7/G).