I know the US has a long tradition of limiting immigration of non-whites with laws like the Chinese Exclusion Act and various quotas (and to a possibly lesser extent, the Alien and Sedition Acts), but when and how have such laws applied to Europeans or targeted Europeans? Would people like my ancestors (Germans who came over in the 1800s) have had full citizenship after passing through Ellis Island or was there more they needed to do in subsequent years? When did the immigration process start become the complicated mess it is today?
The US has always had immigration laws and has always turned some immigrants away, usually depending on what the American economy was like at the time.
There were a number of reasons that people like your ancestors were turned away in NYC, they mostly fell under two categories: 1 - is this person dangerous? 2 - will this person be a burden? If the answer to either of those questions was a likely "yes" (readily communicable diseases fell into the "threat" category) they would have been shipped back.
Per the time to full citizenship, I believe that one had to reside in the country and not break any laws/support themselves for some period of time. I believe that it was 3 years.