When I read CVs I mostly care about the general things people are familiar with. For example, I don't care that you've used GATK or the tuxedo suite, but that you're familiar with and have experience in variant calling and RNAseq. The same goes with databases.

For programming languages, however, it's good to list the ones you're familiar with. You can omit "general applications" (and "operating systems" if you have enough experience that it's obvious that you know your way around the CLI).

When I receive a CV I read the skills and the expertise and compare to my requirements. However, I found often people say: "I know Blast" but what they mean is "I used NCBI Blast interface to run a query once" which is completely different to "I build a large and complex annotation pipeline using Blast+ and other tools".

I prefer to suggest to put the tools as part of the main use, like:" I participate in the genome assembly of the big bird, using reads from deep coverage with Illumina data, I used a large SGE cluster and parallelize SOAP-denovo assembly using MPI", which is more descriptive and informative than just listing "SOAP-denovo".

1) Is it a good idea to put all/most programs onto my resume? Do employers want to see all?

I would list only the skills/programs relevant to the job you are applying for. This implies to make a different CV for each application.

2) Is it a good idea to organize different software/tools into different categories? If so, how would you group them?

Yes, everything that improves readibility is a good idea. The way you group them makes sense to me.

If you want to pass through the HR manager's list of requirements which he/she rarely understands how much time it takes to master only a single of those items in that list, surely you can write down all the languages and tech stack you have experience with.

I would reformulate the question though: "What projects should I put on my resume?"

In my view links to your work/github/website tells much more about a list of languages. The reason I am underlining this is because there is a feed forward loop between candidates and HR managers. HR managers inflate the language requirements, in return candidates inflate their list of languages and the same loop goes on, practically to the moon..