Is there a way to completely disable all Bixby functionality without rooting the S9+? I would like to get rid of Bixby altogether and preferably reassign the Bixby hardware button to a different function.

Is that at all possible and if so: how?
This article has a useful guide to disabling bixby.

Disable Bixby Voice
Go to Bixby home. Click the three vertical dots then settings. Uncheck Bixby Voice
Disable Bixby Button
Go to Bixby home. Click the cog. Uncheck Bixby Key
Remove Bixby from Home screen
From home screen, hold down on empty space. Swipe right until you see Bixby home screen. Uncheck Bixby Home.

This other article covers remapping Bixby button. You will have to keep the button enabled. Skip step two in the previous article.
This one is longer, but basically you download an app that will remap the key to what ever action that can be done.
EDIT
As of Android Pie (9.0), this guide is no longer accurate to disable Bixby. The Bixby button can still be remapped using bxActions(free) or Tasker beta(paid)(thanks beeshyams) and Bixby Home can be removed from the home screen. However, disabling Bixby completely will most likely require removing via ADB or possibly using Package Disabler Pro+(Paid app).