Yes it is safe to uninstall AOSP keyboard.
I kinda feel uncomfortable writing a one lined answer... So I'll explain things a bit more in detail.
Basically, you can remove any app from android system as far as its not used by critical system functions. I dont not know of any specific up to date list that contains all apps that can be removed. But consider it this way:
What can a keyboard app be required for? Can it be used by some other app? The answer is pretty simple. The keyboard app only inputs user data. So if you have alternative, its pretty safe to remove. Even if you dont put any keyboard app at all, your system will work just fine even then. It will just get a little trickier to input stuff.
Again, precaution is always better than cure. So keep that in mind and backup first ;)
For the second part:
Correct me if I am mistaken, but thats a general thing in android. If you are not running an application does not mean that system is not running it. There could always be background services which can trigger some activity from an application. Especially for something as common as input methods, they generally stay in memory so they can used as soon as required.
So to keep an app dead, you must make sure that no background service ever calls it.