What does the padj really mean? I believe it's a BH method, where it attempts to quantify the likelihood of a "false positive" but I am not sure. Also, what does it mean if the resulting calculations nearly all have the same padj values despite having different p-values?
If we look at documentation of DESEQ2 and search "adjusted p-value", we find the section "Multiple test correction". In this section they discuss Benjamini-Hochberg ("BH") false discovery rate (FDR) correction procedures:
FDR/Benjamini-Hochberg: Benjamini and Hochberg (1995) defined the concept of FDR and created an algorithm to control the expected FDR below a specified level given a list of independent p-values. An interpretation of the BH method for controlling the FDR is implemented in DESeq2 in which we rank the genes by p-value, then multiply each ranked p-value by m/rank.
Earlier, they write:
m = total number of tests
They further add:
So what does FDR < 0.05 mean? By setting the FDR cutoff to < 0.05, we’re saying that the proportion of false positives we expect amongst our differentially expressed genes is 5%. For example, if you call 500 genes as differentially expressed with an FDR cutoff of 0.05, you expect 25 of them to be false positives.
I would suggest reading Benjamini and Hochberg (1995) to learn more details if desired.