So I'm about to install the Sims 3 after a long break. I originally installed the Sims 3 & Showtime on two separate disks.
I lost the Showtime disk (which is ironically part of the reason why I went on a break) but I registered both games and recently I found out I can download them from Origin.
Will I have any problems if I installed the Sims 3 CD and the Showtime digital download? Will I need any CDs running to launch the game?
To avoid confusion I'm installing the Sims 3 on a completely different computer than I originally installed the game on (I uninstalled the game a long time ago and recently had to have the hard drive replaced)

No problems. Most of my EP's are from origin and I play without a disk, starting the game straight from TS3W.EXE shortcut without login in to Origin. Of course, I have the release disc version of the base game and if I remember right, the post 2012 base game versions require login in to Origin to play. But yeah, you can mix cd and digital versions.

If you registered your CDROMs with Origin/EA... you'll as mentioned be able to just download all of the content digitally. Otherwise install the CDROMs first... then PATCH until it is up to date (may take more than once)... then install the downloadable content expansions/stuff-packs... and finally once all of the above is patched... you can install CC/mods/whatever else. The reason it needs to be done in that order is to make sure your GameplayData.package got patched properly as it is basically the scripting heart of the game.

I bought the first TS3 but everything else (about 12 other upgrades/stuff packs) was digital through Origin. They all work fine as far as I can tell.