TL;DR: No.
The Yongnuo YN-560IV is a "manual-only" flash on a camera hotshoe. If you look at the foot of the flash, it only has a single pin--the one that communicates sync/fire. If you look at the foot of your D5500, it has four contacts. Those other three contacts are how the flash communicates with the camera body to perform other functions like TTL and HSS.
The reason this flash is so cheap is that it doesn't do all that stuff. :)
... I'm wondering if I made the wrong decision...
Probably. :) You may want to see What features should one look for when selecting a flash?
However,
Firstly, in "Auto" mode the popup flash continually tried to "pop-up" underneath the YN-560…
This is my opinion, but if you want to learn flash, your first step is to get comfortably shooting in M mode. Auto is the very last shooting mode you want to be in. You should be shooting in the PSAM modes, at the very least, and should be comfortable swapping stops between iso, aperture, and shutter speed in your head.
This is because flash requires that you relearn everything you think you know about exposure and light. And your meter, while it's great at measuring the light that's in the scene, can't measure what isn't there. And the flash burst isn't there, yet. TTL manages to fix this a little, by creating a meterable pre-flash burst before taking the shot, but exposure isn't just about getting the needle to "0" any more.
It's also about the ratio of the light between the ambient (every bit of light that isn't from the flash) and the flash. And the only way to really control that precisely is to be in M mode.
Next up, in aperture priority mode, I get a really slow shutter speed, as if the flash weren't there, I know that it doesn't know how bright the flash will be, but surely it knows it's there!? I can always adjust the flash brightness if needed after that point...
...And this is the other reason you want to get comfortable with being in M. The way the A and S modes are programmed makes an on-camera flash very useful for fill (i.e., balancing mostly for ambient, with a bit of flash to "fill in" the shadows), but behaves in a real weird way as main illumination in low-light. If you don't want to have to learn flash/ambient balance, put the camera in "P" mode. There may also be a custom function in your camera to set the shutter speed to be 1/200s (your sync speed) in A mode. This will eliminate the problem in low light.
So is the only way to use this flash successfully to put the camera in manual mode and do it all, manually? ie. Did I make a foolish choice buying this flash!?
With a manual-only flash, you only have M mode on the flash. On the camera, you can use it with any shooting mode, as long as you're prepared to adjust the power output directly on the flash for what you need.
The choice was foolish only if you want to rely on automated modes, and use the flash on-camera all the time. The reason that model is recommended so often, is that it does things OEM flashes don't do as an off-camera flash in a studio-style lighting setup. In that type of setup, most folks prefer manual for precision and consistency and rarely use TTL. A wedding shooter who has to get stuff on the hoof run'n'gun style, otoh, depends on TTL. It all depends on what/how you want to shoot. And manual-only gear is a lot less expensive than TTL-capable gear.
I would say, however, that if you want a cheap flash that can automate its power output, you have two choices. Look for a flash with Nikon TTL capability, such as a Godox TT685/V860II or Yongnuo YN-586EX/685. Or look for an older Nikon flash with AUTO mode, e.g., SB-26, or a used SB-700/SB-800 (some current Metz units have this, too. Wouldn't recommend looking for a current Vivitar that does. And be careful of sync voltage, though, if you go looking at vintage flashes. Your D5500 can only take 0-250V on the hotshoe. Some vintage flashes might have negative polarity or go over 300V for sync voltage).
AUTO (tends to go with SU-4 mode on Nikon speedlights) is the technology that was used to automate flash power output before TTL. It uses a sensor on the flash to cut off the power. It does require that you set the iso and aperture settings you're using on the flash as well as the camera, but the flash doesn't have to talk to the camera for this mode to work. See also, the Strobist on shooting events without TTL. Why Yongnuo/Godox don't put an autothyristor mode into their manual-only flashes still stumps me.
