I use Jrummy Rootchecker..
You can download the apk without play store and just sideload it from SD.
use APK Downloader
just put the apps id in the box and generate download link, and you can download the apk direct.
The Jrummy rootchecker address is com.jrummyapps.rootchecker
For busybox, again just install busybox installer from play store (or download apk as above) id is com.jrummy.busybox.installer - then run it and follow the instructions to install busybox
For terminal - use terminal emulator - id is jackpal.androidterm
all three of those working fine with me and i have full root, busybox and terminal.
the busybox installer will automatically check root, and download the applicable busybox for you, terminal installs as a normal apk. but you dont NEED them as long as your root is working. some applications do need busybox, but you dont have to have terminal.
Bit weird that I just did a factory reset and it rebooted to CWM and then when I rebooted, it hasn't resetted at all!
Still not sure why Phone Storage shows 4GB total but Internal Storage only shows 1.55GB total. Is 2.45GB used for the ROM?