1) Apply FDR to all p-values
2) This happens often; the FDR "q" value is often the same for many of the results, and it does get high pretty fast. I'm not sure why it happens. I've never dealt with 11,000 p-values...
3) Use .05 divided by number of hypothesis results, incrementally increasing to .05. Let's say, for ease of calculation, that you had 10,000 hypothesis results. The cut-off for the #1 result would be .05/10,000, which equals .000005. The cut-off for the #2 result would be twice that, which equals .000010, the cut-off for the #3 result would be thrice that, which equals .000015, ..., the cut-off for the #10,000 result would be 10,000-times that, which equals .05
Here's a little calculator, so you can see some typical results. Notice that there are a lot of similar q-values in this example too.
FDR Calculator in Excel