Haxchi and the web hack are both means of kicking off some code, e.g. Homebrew Launcher, so in that sense they're kind of the same. One difference though, is that Haxchi lives inside the content of an installed ds game, so once Haxchi is installed, it'll be available to you just like that game would have been. It'll be on the WiiU menu, and it doesn't need an active network connection in order to run. However, like most other titles, it has to be executed when (and each time) you start the WiiU.
While it's likely that some easily installable NUS titles (ones that we could install using disc-sourced tickets) could be exploited with a content hack, this particular hack uses games whose legitimate tickets can currently only come from eshop. Keep in mind that you just need to exploit some vulnerability to get your first bit of code running, and Haxchi takes advantage of convenient features of the VC DS implementation to do that.
The idea of using redNAND is to be working from a copy so that if you screw up your copy, you'll still have your original. Plus, since you're initializing using patched code, the patches are there before the system encounters any titles or modifications which would rely on them. The thing is, we've found that as long as you don't do stupid things, the stock 5.5.1 WiiU (without the patches being active) doesn't care much about illegitimate code until you try to run it. Most people are happy to install their fake-ticketed games and DLC on their real NAND, so they choose to skip having redNAND, and simply execute the patches after the system is up and running.
Whether you're talking about loading a pre-modified image, or applying the patches 'live', yes, you need to run it each time you boot. Even if you convince the WiiU to run it for you (coldboothax), or simplify its execution (haxchi), you have to get it to run.
It's not emuNAND. Signature checks and a couple other things are patched, but you're making all of your installs/deletes/saves and everything else right on your real NAND.