Earth formed from heavy elements produced in at least one prior generation star, but there could be more than just one prior generation (It's very, very unlikely that all or nearly all the medium weight elements above lithium, in our solar system, came from just one older star, and pretty unlikely even that all the heavies above Iron were cooked up in just one supernova).
It's not a safe assumption that stars last an average of 10 Billion years. The most numerous types, red dwarfs, make up 80-90% of all stars, last a lot longer than that, and probably stay stable on the main sequence for 100-200 billion years (American Billions). They also shouldn't spread elements around much when they finally do leave the main sequence. Stars about the size of our Sun, spectral class G2, typically live about 10 Billion years, but make up only about 2% of stars. Big stars, type O, B, and A, burn more quickly, and it's possible to get enough hydrogen together for a star to burn through all its fuel and supernova in mere hundreds of thousands of years, or possibly even a blazing fast 10's of thousands. Those stars are rare, but they are so massive that even a few produce enough heavy elements and push enough gas around when they supernova, to create hundreds of sun sized and smaller stars and all the heavy elements to give such stars the solid, rocky planets we now think are practically ubiquitous.
The supernova explosions are a common source for two effects - heavy element formation, and compressive shock waves that trigger new stars forming in nearby interstellar gas clouds. Many of these gas clouds are already enriched with heavy metals from previous supernovae, Spiral galaxies tend to get regions of new star formation, and quiet regions. But, the high and low density regions in spirals like our Milky Way exist on larger scales than the star forming "nursery" clouds, and this is largely because gas clouds are not just compressed by novae - both the dense star forming clouds and very large but more difuse clouds colide with other clouds, including clouds that were part of dwarf galaxies being captured by the big spirals. So, it's a partial coincidence - Older generation stars have some influence on the shapes of spiral galaxy features, but dwarf galaxy capture has more, and the rare colisions of spirals with other big galaxies show just how much influence the large scale objects can have, producing wildly twisted galaxys such as
If anyone wants to read up on this sort of thing, please remember, because astronomers named them before they knew anything about why there were multiple distinct types of stars in the same mass ranges, Population II stars are actually older than Population I, and Population III older than II. A given population usually includes multiple generations of stars. As an exception, the very oldest, massive stars that novaed within the first million years or so after star formation began, and produced so many heavy elements are called Population III, and most probably represent just a single generation and possibly only the largest types.
and, for those people wanting more than just the Wikipedia versions, a little real source material: