I am not sure about all the uncertainties, but I'll try here.

First, nothing really changed since last summer when the obliquity handling was first introduced except for more warning messages recently.

3drefit doesn't actually do any transformations. That's what the scary messages were trying to get across. It only removes the oblique transformation from the AFNI header. On a NIFTI file, this doesn't work because NIFTI has its own header. You could get around this particular problem by using 3dcopy to copy the NIFTI file to AFNI format if you wanted. As the warning message says, if you really want to deoblique, you must use 3dWarp.

The time shifting of the slices will not work properly unless you do it before 3dWarp. How noticeable that will be depends on the amount of the rotation and the order the slices are acquired. The time shifting is slice dependent, but once you rotate the data, the original slice that corresponded to each voxel is lost. (It could be calculated, of course, but that makes life too complicated.)

There are good reasons to acquire oblique slices, but it does make processing a little more complicated.