Should I use every Thunderhead buff I get?

No, prioritize refreshing the DoT when it is falling off (less than three seconds remaining). This means that you will not use every Thunderhead buff, but since most of the damage is on the DoT portion, efficient upkeep on the DoT is more important than maximizing Thunderhead usage.

How do I use Umbral Soul?

Umbral Soul is a downtime spell, used to build up to and maintain Umbral Ice III with three Umbral Hearts as well as get back to full MP. When you can actively hit a target, you should not be using Umbral Soul. Since it can only be used in Umbral Ice, you may need to Transpose from Astral Fire before you can use it.

Is it okay to refresh High Thunder early if I need it for movement?

If you have no other movement options in the given situation, using the proc early is better than losing out on the GCD entirely. However, try to plan better for next time you’re in the situation to have something available for movement, allowing you to refresh the DoT with more proper timing.

Do I do anything to align with raid buffs/maximize damage dealt within raid buffs?

Often it will be advantageous to align pot uses with the rest of the party. oGCD abilities if used roughly on cd will often keep some semblance of alignment naturally with other buffs. If you can spare any polyglot stacks for Xenoglossy for use in raid buffs that you won’t need elsewhere to help keep uptime, it’s a pretty reasonable consideration to do so. Otherwise, BLM will generally fit good spells within raid buffs regardless of alignment–prioritize uptime and solid rotational choices first and foremost.

Should I use Paradox in Umbral Ice even if I don't need it for weaving/movement?

Yes, Paradox is a strong spell and should generally be used every Umbral Ice cycle.