Density

Normal planet density in Stars! is approximately one planet for every 5000 l.y.^2. That is, if you divided the universe up into squares 70 l.y.^2. on each side, there would on average be one planet in each square.

Selecting Sparse decreases density to approximately one planet for every 6500 l.y.^2.
Selecting Dense increases the density to approximately one planet for every 4000 l.y.^2.
Choosing Packed increases the density in number of planets by 30% over Dense in all but the Huge universe (which is stuffed as full as is playable by the Dense setting).

How many and how dense are the stars in various universe setups?

Here is the number of planets / density information for all universes, credited to Leonard Dickens.

Number of planets in a galaxy.

These numbers are not exact, actual number of planets may vary by 1 in a tiny/sparse, or 6+ in a huge/packed

Number of stars

     Sparse  Normal Dense  Packed
 Tiny    24      32     40     60
 Small   96     128    160    240
 Medium 216     288    360    540
 Large  384     512    640    910
 Huge   600     800    940    945

Size

       Edge(ly) Area(ly^2)
 Tiny     400        160K
 Small    800        640K
 Medium  1200       1440K
 Large   1600       2560K
 Huge    2000       4000K

Density of stars (#stars/10000ly^2)

      Sparse  Normal Dense  Packed
T/S/M    1.5    2.00  2.50  3.75
Large    1.5    2.00  2.50  3.55
Huge     1.5    2.00  2.35  2.36