This is a bit misleading. A market maker is never required to buy or sell stock at the “current” price, he always sets the prices at which he is willing to trade.
A market maker is contractually obligated to maintain liquidity - that is, he must always be willing to both buy and sell stock at some price. The obligation is specified as the bid-ask spread he must always maintain, i.e. the difference between buy and sell prices.
For example, suppose the obligation for the market maker in widget-maker WXX is to maintain a maximum 1% bid-ask spread in 1,000 shares. His current bid-ask might be 120/121. This means that he is currently willing to buy at least 1,000 shares of WXX at 120 or sell 1,000 shares at 121, but only so long as this quote is current. If news of a widget glut hits the wire, the market-maker would instantly react to that news, he might immediately change his quote to 115/116 - hopefully before anyone has time to sell any shares to him at $120. He has no obligation to buy any stock between 120 and 115 unless someone executes a trade with him before he changes his quote. But if his current quote is that he’s now willing to pay only $115 to buy stock, he does have an obligation that he must also be willing to sell stock at a price no more than 1% higher than the new bid, i.e. at around $116, i.e. he must maintain the bid-ask spread of no more than 1%.
The art of market making is to discount news, and to react to the general ebb and flow of supply and demand, in such a way that buyers and sellers roughly balance out at the current quoted bid-ask. In the above example, let’s say the $5 drop in price to discount the news of the widget glut is correctly judged: perhaps some people are still panicked at the news and will sell WXX at $115; while others think WXX is now cheap and come in to buy at $116. If the market maker gets it right like this, he may do business on both the bid and ask sides of his quote, making a profit on his bid-ask spread. But if he gets it wrong - let’s say the news is more negative than he perceives - then he may still see only sellers at his new bid of $115. In this case, after buying some stock at $115 he might react again to drop his price further, let’s say now 112/113, and he would have lost money on any stock he bought at $115 before dropping his price.
But market makers are never under any obligation to buy or sell stock and any particular price, just to maintain the bid-ask spread. So they are not really a “buyer of last resort”, more a liquidity-provider of last resort. This means that investors can always have confidence that the market will not vanish, there will always be some price available to trade with reasonable transaction costs (bid-ask spread, 1%). Market makers are an important part of the price discovery mechanism.
No, a net short position may involve certain technical steps to maintain, but economically it’s simply the opposite of a long position. Buys and sells net out. A market maker has either a net long or short position at any point in time. If he’s long and he doesn’t like the position he will simply try to sell it; if he’s short and he doesn’t like it he will buy to “cover” his short.
Delta is used by derivates traders to express their market exposure in terms of equivalent number of shares in the cash market. It’s not really applicable to a discussion of market makers in the cash market.