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Simple, install an app that require root access and check if it works.
*Whether your root is full or half, it depends on the method of root you used.
*Unlocking bootloader is only necessary when you need to install any custom rom.
Otherwise rooted phone can function fully with the locked bootloader.
*Framework-res.apk is used to tweak your phone's appearance.
However, manually changing it can mess your phone.
I would suggest that install Xposed Installer, then download modules using xposed installer and you can use these modules to tweak your phone.
Use Towelroot to root your phone. Then run remountfix. This is to rectify the reboot issue. Once run, your rooted phone wont reboot.
Link: http://forum.xda-developers.com/xperia-m/general/towel-root-steps-to-root-t2789990