To give you guys more details.
When i told you the battery was worsening by the day, I have not been accurate. While the battery is indeed old and near death, there may be something else going on to worsen the situation.

Since three or four days ago, if the phone dies completely, I can only charge it from the power socket. Not from my laptop, not from my "power bank", EVEN using the 2.1A USB connector - 2.1A is in theory as much as the phone can take, but clearly the wall power outlet is doing a better job here. When i try to charge it from anything else than the wall power socket, phone wakes up, starts, after few seconds it returns a "battery low" message, then dies again, and it keeps cycling like this indefinitely. Clearly, it started drawing more power at boot. This could be software-related, but there was no update since few months ago, and I did not install any new App - I may have uninstalled a couple, actually. At the same time, the phone also started to randomly die on me, as if task would suddenly start and draw a lot of power.
Notice that both when this happened, AND upon boot, the phone gets really hot. Again, AS IF some CPU intensive task was running, but nothing really is.

So, I am suspecting there's some more serious hardware issue here.