With regards to the New Deal and how the government financed public works projects, it was largely due to a massive set of spending cuts in every government department.

The New Deal wasn't just one specific thing; the term encompasses a massive set of different pieces of legislation passed aimed at bolstering the US economy in a variety of different ways. Some of this legislation centered on the formation and expansion of the Public Works Administration (see the National Industrial Recovery Act which first established the PWA), some centered on artificially adjusting the markets (e.g. the Agricultural Adjustment Act, the National Industrial Recovery Act), as well as acts that increased regulations in matters such as labor rights and agricultural sustainability. In short, the government wasn't only spending money to create jobs and improve markets (the funding for which came from the aforementioned spending cuts) but also regulating and changing the way those markets behaved.

It's important to clarify, as well, that the Great Depression didn't mean everybody was suddenly broke. Many people, especially many in the middle class, were newly impoverished; the upper class, however, didn't suffer the ill effects of the Great Depression nearly as much, on the whole. US Industries did not completely collapse, though it did experience sharp declines in productivity, revenue, and (obviously) investments. One of the government's main focuses was on revitalizing the middle class, which is why the Social Security act was introduced.

As for WWII, the US war effort was financed primarily by a massive tax hike (the Victory tax, first instituted in 1942), and public investment in war bonds. The US government made a point to advertise the purchasing of war bonds as a patriotic act and a way of aiding the war effort, and created big, nationwide campaigns, usually with endorsements and appearances from war heroes and celebrities, to encourage people to buy them. For anyone reading who might not know, a war bond was basically a way of investing money; the money used to purchase the bond went directly to the government, and the government agreed to repay the bond at a later date, plus interest.