I have University and Nightlife expansion packs and I would like to download the patch to fix some problems from the base game. Am I supposed to download the base game patch or the Nightlife one?

I'm pretty sure you have to download the Nightlife one since that's the latest expansion you have.

download the nightlife patch

You need ALL patches for ALL games; therefore if you have Uni and NL you need the basegame patch, the Uni Patch and the NL patch. However, the NL patch also contains the Uni updates so you only need to download that and the base patch.

Why don't they put the old patches into the expansion packs? Would it damage something if the game were patched twice?

We got the Holiday Edition early this year. We assumed that it came with the patches as well as the Holiday Stuff pack. Did it?

No, not all was included in the EPs- for example, there was a NL patch released after OFB was released. It will not damage your game to patch twice; rather, the installer will tell you that your computer is up to date and not apply the patch if not needed. I believe that the Holiday edition had patches included, and I daresay that this Christmas' Holiday edition will, too. But it is not really possible to include all patches in the EPs as not everyone has all EPs, so they left most as downloadable code.