[Q] Busybox and its interaction with my phone topic

I originally wanted to use Lucky Patcher, main reason why I rooted my phone, howeve, it popped out an error message about BusyBox missing from the system (it still kinda sorta worked, but not really). I've tried installing BusyBox with 5 different installers, some say tahat they can't be installed and I should change a direction (even a non-system partition scares them). One app has installed the busybox, but another one can't see it, neither can Lucky Patcher. Also, no means of making /system/ writeable work (special apps, ES File Explorer, Total Commander, File Commander).

Wat the hell should I do to make my BusyBox installation existent, solid and visible? (and also get an actual ability to make that /system/ writeable).
I'm using Sony Xperia Z1 C6903 on Android 4.4.4, build number 14.4.A.0.157