Delta neutral is probably universal. As far as the rest of the greeks, you have to understand that MMs tend to get the garbage no one else wants. It is then their job to offset that risk as best as possible. Over time, all the good luck and bad luck evens out, and they are left with the profit on the spread.
MMs tend to be more aggressive in making markets on teh side that would "lock" in a profit for them, or bring their greeks more neutral or to their advantage. If you sat there and only made markets so that you could have the best potions on at all times, no one would do business with you. It is mostly risk management, not so much trading. Very different mentality than what you are used to thinking.
Essentially yes. You can parallel shift the curve up/down, rock it around a pivot steeper/shallower, or the angle at which the curve enters the wings.
Not really, although some market makers will sometimes do things based on the client. I know of many instances where a MM took a trade, and almost immediately exited the position he just entered a few seconds earlier at a loss, and went double the direction of what he had just made a market in the direction of the client. This is art, and some traders have a nose for inside information. The information flow is also different in a pit than on the screen. Much much harder on the screen. Option traders trade vola and things that depend on vola. The underlying direction is not what matters, although if you are getting it cheaply, I don't know anyone, MM or otherwise, that doesn't like being long gamma.
As a seat holder you are required to make markets a certain % of the time in the allocations you have. Different size MMs use different strategies, but sometimes you are very wide with no intent on ever being filled, just meeting your regulatory obligation. When you do make tight markets, it is often to offset something in your current inventory - to manage risk. You build a book at first with lots of risk (never delta risk overnight but you may hold it for a bit), and slowly manage risk around it. You are constantly thinking about what is about to expire, and at the same time building your book for the farther out months. Your home runs can be in any of the greeks, but front month positions are where your home runs mostly come from because it is where you tend to try to get long gamma (gamma->infinity as expiration nears.)
Ideally, MMs over the long run make the spread over many trades by working into positions they want, staying as close to delta neutral as possible, and hit a home run here and there because of good luck by being long gamma. In a way, they are trading yield, not price. They get rich because of leverage, size, a steady theta income, and managing that leverage very very carefully. Easy to say, immense skill to do right.
There is always room for game theory. But look, I am a chess player and a bridge player. There is a great deal of game theory here too, but you can become a damn good chess or bridge player never having heard of any of these things. The money is made by understanding how to trade vola with size positions, and manage risk until you get the opportunity to lock that profit in.