Hi Johann,
Apologies for the delay getting back to you.
Yes, genes can overlap between sets with significant and non-significant FDRs, in fact, this overlap can help explain why a set may have a significant P-value but not a significant FDR.
I don't know what type of data you're working with, or how many samples you have in each group so I can't really say if this is appropriate for you, but generally the results from standard mode are superior to preranked if you have enough data to run phenotype permutation testing rather than gene set permutation testing.
There is nothing inherently wrong with looking at potential hits that have a significant P-value but a FDR between 0.05 and 0.25, but remember that these "hits" are likely better explained by other gene sets in the enrichment and confidence in their specific relevance is low. Remember that GSEA, like any kind of pathway or gene set analysis, should be considered a hypothesis generating tool and that biological relevance of the hits should be validated with independent lines of evidence.