Xperia SP looks like it's charging, but it doesn't charge My phone is Xperia SP, it's on CM 12.1, unofficial build (because the official one is buggy due to bad kernel). It has been bought second-hand, so I'm not sure about its history (the phone was in a good state).

So, the problems started gradually, but the symptom is this: when I connect the phone to the charger, the phone goes into it's charging state, the battery icon shows the charging symbol, the LED shines red, but the phone doesn't actually charge, and the battery percent actually actually descends. I would usually just detach the phone, and reconnect it, and it would actually start charging. However, this happened more often as the time passed, to the point where it is completely empty, and it cannot be turned turned on, or charged. When I connect it, the LED goes red (the usual charging behaviour), but then it goes into booting the system, instead of going into "offline charging" mode. It fails to boot the system, because the battery is empty. It then turns off, returns to the red LED state, and from time to time it boots into recovery, which is quite unusual, since by default one has to press volume buttons to enter recovery. While in recovery, usually it would display the battery percentage, but now the place which should show the battery percent is empty.

Are these the symptoms of an faulty battery, or a more severe issue? Did anyone experience anything similar?
I had a similar problem a few years ago with my old HTC EVO 3D which was also running cyanogenmod at this time. I tried flashing different roms and kernels but the problem was persistent. Because my battery was empty as yours is at the moment, I charged it by cutting a USB Cable and applying "+" and "-" to the battery directly. It looked really bad to be honest but it worked and I was able to go back to stock. But that didn't help. The point is - I could make sure that the battery is not defective because I was able to charge it. In the end the problem was the usb port of my phone which had to be replaced. It worked but to make sure my phone does not turn into a torch I also replaced the battery. 

What you describe could be the usb charging port - there might be a faulty contact which prevents the battery from charging but this is just guessing. Bottomline: Try to charge your phone by a usb cable (please be very careful while doing this) or visit a service center.