Pets not showing, problem Sims 4 Cats&Dogs!

You have a few Windows Update failures, Make sure that Windows is updated properly (don't think that has anything to do with your issue but it's worth looking into nonetheless).

have you tried renameing your user folder yet ?

If not, Try  the factory reset described in this post: http://answers.ea.com/t5/The-Sims-4/FAQ-Troubleshooting-Steps-amp-Known-Issues/m-p/3621199

You won't loose anything. Just make sure to not delete any files  and make a backup.

After renaming, repair your game by right clicking on the game in Origin.

Try on a new game with nothing added back first (start new game). If it works, put back your save game and try again. If it still works, you can start putting stuff back.

• Options.ini (or just re-set all your settings manually)
• Tray Folder
• Screenshots/Videos/Custom music
• Rest of the save games (leave out slot_00000001.save which is the autosave and no longer needed - if you still have it)
• Custom content like clothing, hair, skins, make up ....
• Mods (one by one, making sure they all still work)

The rest does not need putting back, cause they are mostly cache files that recreate or log files that are not needed for the game.