Yes, the Wehrmacht soldiers were drafted. A law was passed 1935. The length of duty was 1 year at first, 2 years since 1936. Basic training of 8 to 12 weeks.

Source: http://www.lexikon-der-wehrmacht.de/Soldat/Wehrdienst.htm

Yes, as they needed men to fight as did the US. As memories of 'The Great War' run as a constant reminder of the bloody nature of modern conflict. All nations at this point required a sort of conscription as a result of the overwhelming anti-war agenda that had bult upp as a result of The First World War.

So yes, they had a draft, and so did everyone else, for the exact same reason.

Oh yes, definitely. Nazi Germany was an extremely militarized society, and service was seen as an honor and duty to all male citizens.

As a rule of thumb, pretty much every participating country had conscription in WW2. There was no other way to raise massive armies of millions of people without it. Even Canada had one million soldiers in its army.