What makes you think it’s not really closed?
Just because it shows the Force Stop button doesn’t mean it’s running, merely that at least one of its components is loaded. That can be just about anything. I have apps I know for a fact cannot run in the background that shows the force stop button.
Mainly, it boils down to battery management and the Android architecture. Android apps are very modular, so the Java class for handling push notifications might be loaded but none of its screens or other services would be loaded and it uses negligible amounts of memory. It’s way more battery efficient than reloading it from storage, and if the system needs memory it’ll clear some caches.
What distro I’m using isn’t that helpful of a question because it’s largely a matter of taste and technical needs. I use Arch in large part because I do some rather exotic things that would be harder to set up on most mainstream distros whereas Arch just gives me a completely blank slate to work with and configure my system the exact way I want it to work. My desktop also has some server duties, it runs VMs, it has multiple GPUs and also drives my TV room independently of my main workstation area.
I usually recommend whichever distro gets you the closest to having everything the way you like out of the box as a starting point just because it’s less frustrating when most things works out of the box. The Arch experience is nothing works out of the box because it doesn’t even come with a box. Arch isn’t necessarily a bad choice even for beginners, but the learning curve is much steeper as a result and some people do like to just learn everything whereas some others prefer to start with the shallow part of the pool rather than diving it headfirst. It’s not like you have to commit to any distribution forever, you can start with something simple to use, learn your way around Linux and then you can upgrade to another distribution as your needs and wants evolves.