How can I turn off all the Steam Inventory/Trading Cards/Community stuff?
Solution 1:
To piggyback on what @Nitrous Oxide (Awesome name) Said... You cannot stop/turn off the Steam Trading Card system. They are earned by playing games you have that has the Trading Card System enabled and random events.
I understand it can be quite frustrating at times to receive two-four cards at once and have to view-acknowledge them in order to make the notification disappear BUT you could simply ignore the entire system and horde cards for some time. Eventually you could trade in sets of them for new quirky badges, backgrounds, and experience for your Steam Profile. OR you could "bite the bullet" and enter the Trading Card community once a month or so and sell all the cards you have for pennies on the dollar to eventually be able to buy something you want.
I hope this helps out a bit.
Solution 2:
Unfortunately, no. The Steam trading cards are rewarded to you when you spend a specific amount of time in a game. If you really want to get rid of the cards put them on the community market or trade them to someone who wants them.