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If I then swap all of those upgrades with Red Heat, I can then choose the order again. From this I play a Multi-Mission Gear and play a Brainstorm, do I need to finish prior triggers before Brainstorm, or can I still choose the order?

If you swap upgrades with Red Heat, you can have a lot of triggers, including another Cog trigger. You choose to resolve each of these triggers one at a time. If you choose Multi-Mission Gears trigger to play Brainstorm, you now must finish Brainstorms effect before continuing with your remaining triggers caused by Red Heats swap.
Source: Wizards of the Coast (2019-10-11)    【Link to Ruling】
Tags: Sergeant Cog, Artillery • Mechanic, Private Red Heat, Infantry • Emergency Response