Some time ago(a few months) I've bought a Galaxy Pocket Neo from a local internet store, which isn't really famous (actually everyone I asked knew nothing about it). It turned out to be a branch of mobileshop.eu, which was also unheard of by me or anyone I know. I'm saying this because I cannot be sure whether the OS itself was bought infected or with a backdoor or something like that.
After several weeks of using, I've noticed rapid battery drain overnight, from 70% to emptying(the phone was turned off when I woke up). Not sure whether it's important, but sometimes, when I turned it on again, I got report that battery is on 252% or similar numbers - I assumed wrong calibration. The weird part is, when my phone 'survived' the night (happened a few times), and I tried to use it in the morning, it felt like I was fighting with someone to control the phone, like someone was tapping on my screen - except no one was near me. I solved that problem once by turning it off (took me a couple of minutes, the other one was fast), and when I figured out I could just remove the battery I used that 'technique' (duh, mornings). Then I'd just turn it back on and it'd be normal, for the rest of the day. After a few days I've started turning off wi-fi&sync (mobile data was always off), and my battery only have lost a couple of percent. When I turn them back on in the morning, it's normal.
The thing that persisted is the silencing of the phone and/or putting it on vibration. I'm sure that didn't press the button accidentally or something, that can happen even when the phone is just sitting on the desk. From what I've noticed, the screen doesn't turn on, but I hear the vibration. Cannot tell whether that happens overnight, since my phone is usually on vibration/silenced during that period of the day. I've checked the app permissions, didn't notice anything weird (if you could instruct me for what to look, I'd be grateful). Only non-play app I've installed is the Youtube Downloader from github.
So, does anyone have any idea what I'm dealing with? I'm quite new smartphone and Android user, but I know my way around desktop PCs/Linux/Java/some C and experimenting with them, so any help is welcome. If it is a virus, what is the best way to remove it? From what I've read, the Android antiviruses aren't really effective (and if there are any that could help me, please list them), and custom ROMs should erase everything (I'm ready to take that step if needed), but are there any more options? Rooting the phone and uninstalling everything? If it's a bug or something I'm missing in settings, please notice me, preferably with instructions how to fix it.