How much do people actually calculate?

On the one hand, this is a skill developed by practice. I've definitely improved over time, and get out of practice if I don't play chess for a while. On the other hand, like music, sports, learning a language, or pretty much anything, there's an element of talent involved.

I'd say just work at it. It takes a lot of time, but it certainly can be improved.

10-12 moves is certainly possible for top players, but is not needed at your level. Focus on getting just a few moves right at a time. Play some correspondence chess. First try to calculate in your head. Then use the analysis board to go deeper (and to see the positions more clearly).

As for how much people calculate, I've noticed that sometimes people calculate a lot, but often players (myself included) move without calculating hardly anything, and think they calculate more than they actually do.

A critical component is to know WHEN to calculate. Unfortunately, often times the only way to know this is from experience and knowledge in the position.

Don't confuse knowledge of a position with pure calculation. I play my white opening with almost no calculation, even to the endgame, in most cases, because I'm so comfortable with the setups and I've already studied the most common attacks and plans. Note this does not mean just tactics studying, but what happens is that you start getting a feel when you think the opponent makes a 'fishy' move and start calculating.

Pure calculational skills are always useful, but you'll be surprised at how much less you need to calculate once you start studying actual openings or common positions and plans from the openings you play. Gambit opening are particularly notorious for this - there are some gambit lines that are so risky that GMs won't even go into them over the board because one false move and you're dead. 

Warning: long-winded post!

Calculation depends on several abilities, and you could be weak at one of them but strong at another, but your calculation is only as strong as the weakest of these links.

1) Visualization: the ability to imagine positions in the future.  This can definitely be improved with lots of slow chess (I would recommend denying yourself the analysis board correspondence chess to train this).  Blindfold games are played on this ability alone, and this is a skill that can be trained.  There is much more to calculation than this, though. 

2) Board vision: the ability to see all the possible moves in a position (actual or visualized).  If you've ever been surprised by an opponent's move because you didn't see it, that was a board vision error.  For many (like me) the most often overlooked moves are long moves and backward moves. I think this is more important.  You can visualize 20 moves deep if you like, but with poor board vision you could miss the opponent's response on the very first move you visualize.  You can train board vision, too, but it's tough.

3. Tactical pattern recognition.  This is how well do you know the most common tactical sequences that mate, win material, etc.  Without this you might not consider certain lines because knowing the pattern is the only way you would even consider the move that begins the sequence.  This is probably the easiest one to train, and it is certainly important, as it can sometimes even substitute for hard calculation.

When you really sit down and calculate, you're dangling by a cord of these three strands, and you want the weakest strand to be pretty strong.

In terms of how much I calculate, I think I do it basically every move, at least as soon as I'm out of my opening book knowledge.  I probably do it too much, and too often, frequently putting myself in time trouble.  But I think this is the best way to improve, and that the masters who say they don't calculate much are just unaware of how much they have developed these skills and how they are working all three automatically.  The board vision and pattern recognition especially are operating at an unconscious level as you visualize.

(I have to cite Dan Heisman here for identifying the three types of chess vision--very helpful I think.)

This game is an excellent example of how playing forcing moves makes calculating lines easier.  Since forcing moves are concrete lines they are easier to calculate.  Obviously this takes practice, but it does get easier :-)