So apparently upgrading a set of the solstice armor on one character will double the progression on subsequent characters: see here.

The caveat to this is that it's bugged on Titans, so do your Titan last.

So I haven't done much yet outside of picking up the armor. I'm probably only interested in the titan armor. Maybe just the helmet. Once I upgrade it, can I get it to drop with different rolls? If so, from where? And if I want it as an ornament, do I have to upgrade the entire set and buy it all for 6k bright dust? No option for just the helmet?

So I haven't done much yet outside of picking up the armor. I'm probably only interested in the titan armor. Maybe just the helmet. Once I upgrade it, can I get it to drop with different rolls? If so, from where? And if I want it as an ornament, do I have to upgrade the entire set and buy it all for 6k bright dust? No option for just the helmet?

You can't just upgrade the helmet. You need to complete all objectives for each armor piece, and then you upgrade the entire set of blue armor to purple armor.

Here's some more detail about the various bugs and poorly explained things: here.

It includes a suggested optimum sequence for upgrading your armor on all 3 characters.

From da TWAB;

SOLSTICE BOUNTIES AND BRIGHT DUST
Shortly after Solstice of Heroes launched, Destiny Player Support became aware of an issue where weekly and repeatable bounties for the event were not giving Bright Dust as intended. We are currently investigating solutions to this issue. We are also investigating a solution for those who missed out on earning Bright Dust from these bounties during this time and will be granting Bright Dust to affected players as soon as we can. You can continue to complete and redeem your Solstice of Heroes bounties normally.

KNOWN ISSUES
While we continue investigating various known issues, here is a list of the latest issues that were reported to us in our #Help Forum:

• We are aware of an issue where Gambit Prime will not count for Solstice armor with >Gambit objectives.
  
• We are aware of an issue where Nightfall: The Ordeals will not count for >Solstice armor with strike objectives.
  
• We are investigating reports of doubled progression progress on Solstice >objectives not working for some players.
  
• We are aware of an issue where the Majestic Solstice Bond for Warlocks has an >>objective that incorrectly states to get 150 Guardian Super kills. The objective >progresses correctly with Arc Super kills against any enemy.
  
• We are investigating an issue where shaders do not properly apply to the >Majestic and Magnificent Robes for Warlocks.
  
• We are investigating an issue where shaders do not properly apply to the >Majestic and Magnificent Cloaks for Hunters.

And da Twitters

We are aware of reports regarding certain class glow ornaments not matching their advertised vibrancy, and are investigating. We will update as soon as we have more information.

— Bungie Help (@BungieHelp) August 13, 2020