Otherwise, I will just install Darkness Rising but will I miss out on some content? Do I download Darkness rising .exe from the official server?

If you have access to a direct download of the complete Darkness Rising package you can use that (it actually contains all previous packages including Trials of Atlantis and Catacombs). The full DR installation sums up to ~4 GB. Just be careful not to download the DR update - that's only ~800 MB and requires a pre-installed Catacombs to run.

Thanks, I'm downloading trials to catacombs setup now. After that I will download Darkness Rising but will I still be able to install ToA after that? Is it any good?

Why would you want to do that? As I said before, once you have installed Darkness Rising you already have installed all older extensions (i.e. basic Dark Age of Camelot, Shrouded Isles, Trials of Atlantis, and Catacombs). Running ToA (older) install on top of DR (newer) could easily ruin your installation!

I couldn't find the full package and when I go to the DR download site, it says it requires catacombs so I'm going to download si to ToA setup then install ToA to catacombs then finally catacombs to darkness rising. But I'm not sure if I download the isles_catacombs engine update here?
http://www.trialsofatlantis.com/downloads/ (step 1) or just the 14 day tree trial one. Thanks.

edit: Also, do I have to update the game after each expansion or what?

I was talking about this little DR installer. But your procedure sounds good, as well. It's not the one described here but should also work. Updating each single expansion is not required. It's enough to do it as a final step after installing everything else.