You know, it'd really help if there were a map or apps that laid all of this out logically
Don't we all know it. The DOT does its best, but its online NYC parking map, while a janky piece of technology is still the simplest way (on desktop) to quickly find out what alternate-side parking looks like for any given street. It gets the job done, but not terribly well.
The best and simplest app for my money is SpotAngels. Available for iOS (with Android support coming soon) it offers a map that's not a nightmare to navigate, sends notifications for when alternate-side parking is in effect and how much time you have left to move your car, and is supported in San Francisco, Los Angeles, Chicago, Boston, and a handful of other cities in addition to NYC. If you're using Android, we'd recommend Smooth Parking, which has an accurate and up-to-date map and times for alternate-side parking in the city and also takes street parking holidays into consideration.