You don't need the Toolbox to side-load apps from apks. The Toolbox is just a convenient way to side-load the 4 Google components you need to get the Google Play Store. Then you can install apps from the Amazon App Store, the Google Play Store, or apk files.
The Toolbox can't actually remove the Amazon bloatware to recover memory unless you root the tablet first - it can only disable and hide the apps without rooting. Various ever-evolving methods of rooting are offered - you just have to check the XDA threads to see if one works currently for your Fire tablet model and firmware version. Keep in mind that some apps won't run on rooted devices, so it's a trade-off.