Not sure what would have a week long battery unless it was never used, but iPhone has find my friends app that will do what you want. It of course isn't that cheap but used would be close.
I'm having the same problem. Mine stops at 72% though. I have found I can pull the battery and reinstall and it will raise the percentage some, but 100% has only been possible on my car charger. Samsung needs to push out an update to fix this. I have to wonder if it isn't to make people leave...
I have both a Note 5 and the IPhone 6s plus. To me the iPhone has a much better battery life. I love the Note, and the stylus, but if I am going to be out all day and not around a chrger, I'll take the 6sp.
Some phones don't have the problem at all, so it doesn't matter what case is on the phone, it's going to work. It's kind of like a screen protector. At least what I'm trying to explain is. If you put a screen protector on your screen, it still works when you touch it, right? But, it won't work...
I don't think it's the cases, or the angle of the finger touching the sensor. I believe it's a ground issue. If your hand isn't touching the metal on the sides of the phone, it isn't going to work. Some really thin cases may get enough ground to make the finger print sensor work, just like one...