Are there estimates of the minimum number of genes required to sustain life?
In what I mean by life here, I don't include viruses.
Depends on what kind of life you want! The Database of Essential Genes lists genes essential for life for a number of species, although I would be wary of some of the numbers; bacterial results are probably more reliable but it lists 118 for humans and 2114 for mice. There are a bunch of different numbers for different E. coli strains, with larger numbers being published earlier, go figure.
It of course varies by species. In mid-August of this year, scientists reported the smallest genome yet: 112kb with only 137 protein-coding genes (predicted). Crazy! That doesn't mean it's the smallest number of genes though, just what we've found. Some folks in 2006 predicted the smallest would be 113kb and 151 genes - pretty close close to the above, all in all.
For something a little more familiar and studied, though, Craig Venter has made synthetic biology his goal. Part of trying to create life is knowing what the bare minimum is. Working with Mycoplasma genitalium, he and his crew knocked out each gene and found that 381 were indeed essential.