So, a long time ago I rooted my device using a Windows app called KingoRoot, and after that my tablet didn't boot. The exact same thing happened with my other device. I did not unlock the bootloader because I didn't even know what it was. The tablet will sorta boot but the OS just refuses to load. Is this device bricked? Is it fixable? If it's any help the device is a Thomson QM-736 running Android 4.4 KitKat. I also have both the ADB and Fastboot installed on my PC, however the tablet will not connect (I assume the charging port is dead) so I don't actually know how those will be helpful. Thanks in advance.
To answer your title, no you don't need to unlock bootloader to root your android device. I rooted my device two months ago, with bootloader still locked.
Unlocking bootloader lets you change to which operating system is launched, install/remove operating systems in your device, etc. While rooting gives you root/admin right INSIDE the operating system (i.e. after the OS has been launched), via an exploit. I also used KingRoot, but I installed the KingRoot apk on my phone and let it do the trick. Had ADB and Fastboot configured on my computer, in case it didn't go right, but I didn't needed them.
Remember, unlocking bootloader will wipe all the user content from your phone, like a factory reset. Plus it will wipe the TA partition (which holds DRM keys that lets you use manufacturer's proprietary technology).
In your case, it seems the rooting process wasn't successful. So you can either unlock the bootloader and then reinstall the android operating system (or install a custom version/fork of android), or perhaps you can get your manufacturer's stock OS and install it using some tool that does not require you to unlock the bootloader. Can't say whether it's available for your device/manufacturer though, but I got XperiaFirm and the android image in (.ftf format) ready for myself if my root failed.