I should mention right now that I am not a lawyer. My interest in these cases is wholly historical, and I can't possibly offer the intricacy of legal theory that an immigration lawyer might be able to offer. However, I believe I can offer some insight about wider trends.

I am going to avoid talking at length about Obama's immigration announcement, because that is a political question is against the rules for discussion. However, I will say that the DOJ has released a legal justification for the action, which cites Reno v. American-Arab Anti-Discrimination Committee as its main legal justifier. Sadly, that case was in 1999, so it is also out of the purview of /r/askhistorians. However, I can talk about how immigration has been handled generally by the the Court in the past century.

You are right that the first century of the United States was pretty boring as far as immigration law goes, and that the Chinese Exclusion Act was the first substantial restriction (btw, the act was in 1882, you were close!). From that arose a few cases, each of which solidified a lot of immigration power in the executive branch. The cases gave the executive a lot of wiggle room in deciding how it was going to implement Congress' immigration laws. I'll discuss a few cases.

Chae Chan Ping v. U.S.

This case arose from an amendment to the original Chinese Exclusion Act, known as the Scott Act. Previously, if a Chinese-American wanted to return to China, they secured a return document that allowed them to return to the U.S. without being bothered by the Exclusion. The Scott Act removed that option, and blocked any Chinese from traveling abroad and returning. Chae Chan Ping left before the Scott Act passed, and brought with him his return document. When he returned, he was refused at the border. He, of course, appealed that decision. The Court sided with the government, citing the need of Congress to be able to protect from foreign aggression. This centralized the ability of the federal government to heavily dictate who could leave and enter the country.

Fong Yue Ting v. U.S.

This case arose from the Geary Act, which introduced requirements that each Chinese-American carry with them documents affirming their lawful right to be in the United States, with punishment of deportation if found without such documentation. The law gave current Chinese residents a single year to get documentation. Many defiantly refused to get registered, Fong Yue Ting being among them. Fong Yue Ting was arrested, and appealed his case to the Supreme Court. The Court held that the sovereign right to deal with entering aliens (such as in Chae Chan Ping) was not substantially different from dealing with internal aliens, and that the government had the right to deport any non-citizen for any reason, even if they had entered the country legally.

Nishimura Ekiu v. U.S.

This one deals with a Japanese immigrant. Nishimura Ekiu had arrived in the U.S. claiming that her husband lived in the U.S. and would pick her up. Immigration officers did not believe her, and refused entrance to the U.S. Nishimura Ekiu appealed. The court found that she did not have the right of due process regarding her admittance to the U.S. More specifically, that the court does not have the ability to review the determinations of immigration officers while reviewing potential immigrants.

These three cases (among others) have combined to create a situation that is pretty unique in U.S. jurisprudence--something referred to as "plenary power," which means that a governing agent has a decision over whether or not to act, and that their decision is absolute, ie not subject to judicial review in the terms of the U.S.

This is, of course, not a complete view of immigration law in the U.S., but I believe it does show how the era of Asian exclusion shaped some of the current policy choices being made in the sphere of immigration.

If you want me to talk about the Exclusion Act in regards to something other than the Supreme Court, just ask.

Lots of this was prior knowledge that I can source if anyone asks, but I did base some of this on

Chin, Gabriel J., Chae Chan Ping and Fong Yue Ting: The Origins of Plenary Power. Immigration Law Sources, David Martin and Peter Schuck, eds., Foundation Press 2005.