Is there any way to avoid being dismounted whilst interacting with items outside of Draenor?
Solution 1:
Updated for Legion expansion:
The blacksmithing item Demonsteel Stirrups allows interacting with items while in the Broken Isles.
For older content this is not possible.
Footnote: Druids in travel form and shamans in ghost wolf (can get mount speed in Broken Isles with Rehgar Earthfury as Bodyguard) can interact with items, this is not the same as being mounted but close.