Fed in theory only control one aspect of interest rate, which is the federal fund rate. The federal funds rate is the interest rate that banks charge each other to borrow or lend excess reserves overnight - this sets the baseline for consumer-financial institution interest rate. High interest rate will increase unemployment, as business are not able to take loan (and risks). High unemployment will then in theory reduce inflation, because when everyone is jobless and poor then the value of goods will also come down. This is why inflation is not a good metric to gauge anything, and the whole system is ass. Metrics like for example general mental health of the population, satisfaction with how things are going, upward economic mobility, and levels of education in children should take precedent over inflation numbers, GDP, and fake unemployment numbers.
I don't know, I thought they are talking about inflation as (usually) defined by CPI (consumer price index), e.g., straight up price of different grocery and everyday household products weighted into a composite index
This is the prevailing logic but it should also be mentioned that only a relatively small fraction of the least few years' inflation is attributable to consumer spending, wages, and employment. A large share is due to what is basically price fixing among the large monopolies, taking profits together rather than undercutting one another. Liberals call it price gouging but this is something that doesn't happen very much under non-monopsony economic conditions. All it takes is one undercutting company to ruin the scheme, after all. Capitalism inherently produces monopsony, of course.