4 of 4 Corrected and improved.

Yes, in principle, under a gold standard, the central bank can buy more gold, or build a mine and mine it, which increases money supply. This lowers the short run interest rates. But the issue is that it typically can't fight the market price of gold: when the price of gold increases beyond the parity, people rush to exchange their paper currency for gold at the bank. This happens until money becomes so scarce that ir regains its value relative to gold.

This is very different from what happens with fiat currency. With fiat currency, the central bank can buy bonds, or gold, but it can even just print money and give it away. ( The act of giving it for free to individuals is called a "helicopter drop". Dropping it from a helicopter is one way to actually do it...)

Even though in a gold standard, the central bank can increase money supply, this is different than buying bonds. This is because since money is backed by gold, it can't loose value relative to gold. In this case, it's really hard for the central bank to lower the value of money if it wants, for example, to stop a deflation. Instead, because bonds are also denominated in the country's currency, making money lose value also makes bonds lose value and vice versa.