The bubble algae you are seeing is probably the gametophyte of Derbesia (formerly classified as Halicsytis). This explains hair algae and bubble algae and why it's spreading so much: It's reproducing sexually in your tank (or at least trying to).
The issue: Hair algae usually spreads through fragmentation, meaning every little hair that settles somewhere can grow into a new algae. You won't be able to catch every single hair, so you might need to find the right herbivores for the job. I have not found scientific evidence that popping the bubble algae is actually spreading it but it's still a possibility, so I advice to remove it carefully by hand if you can.
Regarding dinos:
From what I have experienced myself and seen on R2R dinos can, but don't have to appear in low nutrient environments. If they are a symptom or rather the cause of low nutrient levels is highly disputed. That's why I always suggest trying both sides: First try to bring down nutrients, even if they already appear to be at 0. Manual removal of dinoflagellates, substrate cleaning, rinsing rocks you can take out in the bucket with old water during water changes to remove detritus, etc. If that doesn't work you can still try adding nutrients with DIY phosphate and ammonia solutions. Why chemicals instead of more food?
- Better to control
- Direct effect without delay until food is decayed
- Skimmer and other filters won't be affected as much
- No organics are introduced which might be able to cause more problems along the way.
https://www.reef2reef.com/threads/diy-ammonia-dosing-for-low-nitrate-systems.987087/
https://www.reef2reef.com/threads/diy-phosphate-dosing-recipe.1109919/
If all that doesn't help you can look into silicate dosing to encourage diatom growth. They might be able to smother the dinos if you manage to get their numbers high enough. Many people add bacteria solutions as well, which I'm personally skeptical about, but if nothing works you might as well try it. A UV sterilizer can help in some cases as well. It might be able to kill some algae spores, as well as dinos.
Stay patient and try the low nutrient approach with very frequent tank cleaning for a month or two. Only then switch to increasing nutrients.
I hope this helps...