Thank you for your questions.
1. Full employment addresses the number of jobs created which admittedly, have been significant since 2009. However, the concept of full employment says nothing of the quality of jobs. U.S. Fed Chairwoman Janet Yellen has admitted over the last two days that "job creation has been skewed toward lower-pay sectors". She has also stated that "compensation growth hasn't shown sustained pickup" and "if job market keeps improving, wage growth should rise". These are important clarifications because they suggest that wage inflation is more a hope than a realization from the Fed’s perspective. The Fed also admits that long-term unemployment, which refers to people that have been unemployed for a longer period of time (say over a year), is higher than they would like it to be. This is symptomatic of a mismatch between the workplace skills that are demanded by employers versus the skills that are available by would-be employees. The U.S., in fact, has many job openings at the moment that remain vacant because employers cannot find people with the necessary (mostly technical) skills.
The market's inflation expectations, beyond just wages, have also tumbled – which is worrying central bankers around the world. For example, by our calculations, the market's expectations for the length of time it will take inflation to reach the Fed's 2% inflation target has doubled since the beginning of December 2015 – just prior to the Fed's interest rate hike. The Fed, and other central banks, have repeatedly stated that they expect the drop in oil prices to have transitory effects on economic results because oil producers are hurt by low oil prices. But, oil consumers should benefit from more disposable income that may be spent elsewhere. Thus, the economic impact is "balanced". That has not actually happened as oil consumers have not increased their spending elsewhere despite extra savings at the pumps. To really highlight the impact of low oil prices, the ECB has stated that it is paying close attention and is in the process of reviewing the effects low oil prices have on inflation. The ECB's Chief Economist, Peter Praet, even wrote in January that the "effect of [falling oil prices] is not negligible or temporary".
2. The Fed looks at an array of data so I won't speculate as to why they raised interest rates, although strong jobs numbers undoubtedly played a key role in its decision. However, I would state that a central bank's ability to conduct monetary policy is largely premised on its ability to affect expectations that investors have about the future. By changing expectations about the future, the central bank affects not only current short-term interest rates, but also investors' expectations of where those short-term interest rates will be in the future (i.e. the term structure of interest rates). In order to affect investor’s expectations, a central bank must be credible. By setting the stage and repeatedly stating throughout the first three quarters of 2015 that it would be appropriate to raise short-term interest rates "at some point in 2015", it may be argued that the Fed pinned itself into a credibility corner. If it didn't hike by the end of 2015, its ability to conduct monetary policy in the future would have been seriously compromised.
3. I, too, can't speculate on whether inaction by the Fed in 2008/2009 would have led to a repeat of the 1930s. Funny that you asked that as I just finished reading an article that asked whether the Fed's actions in 2008 avoided a depression or just postponed one. I do not believe that the Fed would blindly continue to raise interest rates if signs of a recession, let alone a depression, emerge. In the absence of a fiscal policy response, the Fed had, in my opinion, no choice but to step in and act. Whether the actions they took were the correct ones is impossible to say.