What are easy examples of citizenship proof and how stringent are they?
They're pretty chilled. For example, on the form where they ask for all the dates of where you lived in the USA, I didn't go back to my birth in 1969 and list precise dates; I just put 1969 - 1972 Austin, Texas, 1972 - 1975 Alexandria, Virginia, etc. When it got to later dates that I could take a reasonable guess at, I added the month. Nobody asked for any proof of any of that.
Then there were the mistakes we made.
For our first child, I forgot to bring our marriage certificate. The person who handled our application did a brief interview with my wife and I, asking questions about our marriage ceremony (Where was it? Did anything unexpected/funny happen? etc) and said she was satisfied we were actually married, so the baby's passport application could go ahead.
For the second child, my wife didn't bring any form of ID they could verify her signature with. (We're normally much more organized than this post would suggest.) The person handling our application said they'd look up the application from baby number 1 to compare signatures, and let us know if there was a problem. We never heard about any issues, and the baby 2's passport and other paperwork arrived a few weeks later.
So while I wouldn't suggest skipping anything important, it's not like the UK immigration process where they're looking for a reason to deny you.
Will the passport be picked up at the embassy or is it a separate entity and will be mailed?
The last time we did it (2015) you paid for a courier service as part of the application process at the embassy, and then they courier it to you a few weeks later. I don't know if this is still the case or not.