Options dealers don't necessarily sell options. They can also buy them. And "writing" an option means to sell it, so "wrote the contract and sold it" is redundant.
Options dealers are not automatically short all of the contracts that they sell. They may already have a long position in that option contract, in which case their sale is a closing sale. Also, I think you may be a bit confused...a short sale of options is not determined based on one's position in the underlying; you can be short an option and long the stock at the same time (for example, an overwrite is a sale of upside calls by someone who is already long the stock).
Delta is the first-order change of the option contract value, with respect to a unit change in the underlying price. I wouldn't refer to it at "velocity" because while I understand what you are intending to communicate (it is a rate), there is another options Greek called "speed," which is the change in gamma with respect to the underlying price. Wikipedia has a really nice description of the various options Greeks. For the truly curious, Sheldon Natenberg wrote a nice book called "Options Volatility and Pricing" that explains the Greeks better than I or Wikipedia can.
Options dealers who are short gamma will hedge by buying stock as it goes up, and selling stock as it goes down. Their demand for (or supply of) the underlying stock will impact the underlying price in a way that pushes the price in whatever direction it has already started to move.
In defense of my word choice: theory and practice are not mutually exclusive. If you want to be excellent at something, you should aim to learn both. BTW I have never taken an academic course on options or finance.
