Hello, when I first installed the game with its expansions, the launcher told me that it wasn't patched up to date. I thought it wasn't too important and ignored it. Since then it stopped telling me about it.

However, later I started using Story Progression mod and this page nicely described how to learn your game version through the launcher:
http://nraas.wikispaces.com/Patch+Level+Compatibility

What I found strange is that my game shows as fully patched and up-to-date.

So I shrugged it all off and started using the mod. And I ran into multitudes of bugs, starting from small stuff like family inventory stopping to function and ending with real game-breaking glitches, after which the only solution was to quit without saving. Even drastic measures like resetting sims or evicting them didn't help.

I think that means my game is NOT patched up to date. Nraas page warns that using the wrong version of their mod means you run into many bugs and glitches.

One good solution would be to manually install the full latest patch.

1) Where can I download the full cumulative game patch to install it? Or, if it doesn't exist, the full collection of all patches, I'd install them one by one. My latest EP is Into The Future.

2) If possible, how else can I look up the game version and the patch version correctly?

Never trust the launcher when it says that your game if fully patched. It's always been buggy like that. Have you tried the Sims 3 Patch Downloader -program? It's shows the current installed versions, as well. It's the one I've always used when needed.
http://modthesims.info/d/409829

For some reason it wouldn't work. I press "download all" and nothing happens. Checked the firewall, the full activity for program is allowed. Maybe the links do not work anymore.

Any other way to find all the patches? What about this link:
http://answers.ea.com/t5/The-Sims-3...more/td-p/67858

The "Get content patch" worked, though, but it is only the store content patch.

That's odd. But yes, you can use that superpatch for all regions.