Yes, you must get the Dragon Armor from the Stout Mines before you can pass the Bronze Beast. There are alternative methods of doing this, though, rather than completing the mines (the least violent of which is standing next to Freyedies' private chamber door and "vanishing" into it, snagging the Dragon Armor there, and then "Vanishing" back to the other side). But then you miss the splendour of the "Burning Caverns"!
Other than that, there is no order to how you should proceed after Serpent Temple before Dragonspire. I would suggest getting your fire spells or stone spells up quite a bit before attempting Boogre Caves, as there are Lilies-a-plenty in there. I think Dragonspire was designed to be the last dungeon in the central area (between Ishad N'ha and Brimloch Roon) so that's why it depends on the completion of other dungeons (Shurugeon Castle, Stout Mines) to complete it. Boogre Caves is an offshoot and can be completed at any time without damage to the story line (although you should complete it before Dragonspire to avoid disruption in the Mage guild quests.)
Have fun!
-Sazerac