Drag and drop may be added in the future. For now, you can right-click on a reminder to move it to another tab. There is a trick, though: It can't be the currently selected reminder. So before right-clicking on the reminder, you should deselect the reminder, either by clicking another reminder, or by clicking in the blank area of the window.
The plugin should not assign reminders to tabs "haphazardly".

If it's doing that, it's a bug -- please let me know if this is the case.
For each type of automatic reminder (for example, crafting reminders) you can specify which tab the reminder will go under, for each character. This is in the options panel (click the big "options" button at the top of the window), under the "Character" tab.
First choose the character from the list at the top. Then you can enable/disable different kinds of automatic reminders, and specify which tabs they go in, and which categories (colors). All of these settings are character-specific, so if you have a different tab for each character (for example), the plugin can always put reminders relating to those characters under their own tabs.
When you manually create a reminder (by clicking the big "plus" button at the top of the window), the new reminder will be created under whichever tab you currently have selected.