I don't know anything about gene expression studies but I do have some interest in multiple inference so I will risk an answer on this part of the question anyway.
Personally, I would not approach the problem in that way. I would adjust the error level in the original studies, compute the new overlap and leave the test at the end alone. If the number of differentially expressed genes (and any other result you are using) is already based on adjusted tests, I would argue that you don't need to do anything.
If you cannot go back to the original data and really do want to adjust the p-value, you can indeed multiply it by the number of tests but I don't see why it should have anything to do with the size of list2. It would make more sense to adjust for the total number of tests performed in both studies (i.e. two times the population). This is going to be brutal, though.
To adjust p-values in R, you can use p.adjust(p), where p is a vector of p-values.
p.adjust(p, method="bonferroni") # Bonferroni method, simple multiplication
p.adjust(p, method="holm") # Holm-Bonferroni method, more powerful than Bonferroni
p.adjust(p, method="BH") # Benjamini-Hochberg
As stated in the help file, there is no reason not to use Holm-Bonferroni over Bonferroni as it also provides strong control of the familywise error rate in any case but is more powerful. Benjamini-Hochberg controls the false discovery rate, which is a less stringent criterion.
Edited after the comment below:
The more I think about the problem, the more I think that a correction for multiple comparisons is unnecessary and inappropriate in this situation. This is where the notion of a “family” of hypotheses kicks in. Your last test isn't quite comparable to all the earlier tests, there is no risk of “capitalizing on chance” or cherry-picking significant results, there is only one test of interest and it's legitimate to use the ordinary error level for this one.
Even if you correct aggressively for the many tests performed before, you would still not be directly addressing the main concern, which is the fact that some of the genes in both lists might have been spuriously detected as differentially expressed. The earlier test results still “stand” and if you want to interpret these results while controlling the family-wise error rate, you still need to correct all of them too.
But if the null hypothesis really is true for all genes, any significant result would be a false positive and you would not expect the same gene to be flagged again in the next sample. Overlap between both lists would therefore happen only by chance and this is exactly what the test based on the hypergeometric distribution is testing. So even if the lists of genes are complete junk, the result of that last test is safe. Intuitively, it seems that anything in-between (a mix of true and false hypotheses) should be fine too.
Maybe someone with more experience in this field might weigh in but I think an adjustment would only become necessary if you want to compare the total number of genes detected or find out which ones are differentially expressed, i.e. if you want to interpret the thousands of individual tests performed in each study.
You absolutely do want to apply a correction. The key idea is identifying significance by chance. As you increase the number of comparisons you increase the number of those that will be significant by chance.
For example, let's take the generic example of doing 100 comparisons using a significance threshold of 0.05. Now, a p-value of 0.05 means there is a 5% chance of getting that result when the null hypothesis is true. Therefore, if you do these 100 comparisons, you would expect to find 5 genes significant just by random chance.
As such, to avoid making these false-positives (Type 1 Errors) we 'correct' the p-value thereby making the test more conservative.
The choice in correction can vary too. Bonferroni is a common correction but if you have 1000's of genes, it is going to be exceedingly unlikely you will find anything significant because it will be so conservative. In that case, you may use the 'FDR' (False Discovery Rate) correction. There is no absolute answer so you need to explore the possibilities and make the best choice and of course report what correction you applied.
EDIT
Regarding you comments below I thought an example can help demonstrate the concept.
Using R, I generate completely random values for 250 genes with two treatments (A and B)
set.seed(8)
df <- data.frame(expression=runif(1000),
gene=rep(paste("gene", seq(250)), 4),
treatment = rep(c("A","A","B","B"), each=250))
I then split the data by each gene and run a t.test comparing between the two groups.
out <- do.call("rbind",
lapply(split(df, df$gene), function(x) t.test(expression~treatment, x)$p.value))
Now, given that this is completely random data there shouldn't be any significant differences and yet when I count how many there are 9 significant genes!!!
length(which(out < 0.05))
[1] 9
Avoiding mistakes like these is the point behind making these corrections. Hopefully this helps clarify for you.
Best Answer
To use the Bonferroni correction, you first have to decide how many hypotheses you are testing. That's the factor by which you divide your statistical significance threshold. If you are considering the hypothesis "drug A alters parameter X more than drug B does" for every possible pair of drugs A and B and for every possible parameter X, that's $mn(n-1)$ hypotheses in total. If $m$ and $n$ are large your significance threshold will be very stringent, so it might be better to focus on a smaller set of hypotheses, or not use the Bonferroni correction at all.