How does the Helm of Brilliance interact with Great Weapon Fighting?

The item Helm of Brilliance includes a feature that reads:

You can use an action and speak a Command Word to cause one weapon you are holding to burst into flames. […] When you hit with an Attack using the blazing weapon, the target takes an extra 1d6 fire damage. (DMG, 173)

Great Weapon Fighting reads:

When you roll a 1 or 2 on a damage die for an attack you make with a melee weapon that you are wielding with two hands, you can reroll the die and must use the new roll, even if the new roll is a 1 or a 2. (PHB, 72)

This question deals with what dice are affected by Great Weapon Fighting and it seems that the the feature was only intended to affect the weapons damage dice not extra dice from features like Sneak Attack or Divine Smite, or spells such as Hex. It seems that all of the weapon’s damage die, however, are affected and so weapons such as Frost Brand have their additional dice modified (1d6 cold damage in the case of Frost Brand).

In the case of Helm of Brilliance‘s flame effect does is the additional 1d6 fire damage affected by Great Weapon Fighting?

On one hand it seems like no because it is clearly a spell like effect rather than a natural part of the weapon. On the other hand the effect clearly modifies the weapon rather than the attack and so the additional damage could be considered a quality of the weapon.

How does the Celestial Brilliance interact with the Deeper Darkness spell?

I am in doubt about what happens when two area effects that influence a place’s light interact with one another.

Celestial Brilliance (Book of Exalted Deeds, p. 94) is a 4th-level spell and reads, in part:

Celestial brilliance brought into an area of magical darkness (or vice versa) is temporarily negated, so that the otherwise prevailing light conditions exist in the overlapping areas of effect.

Celestial brilliance counters or dispels any darkness spell of equal or lower level, such as deeper darkness.

Does this mean that if I cast Deeper Darkness as a 3rd-level spell within Celestial Brilliance‘s area, then both neutralize?

What I do not understand is if it is a matter of spell level or timing (i.e. which spell is cast first).

Also, is there any difference – other than the damage – between the light produced by Celestial Brilliance and Daylight? The first one says “light brighter than bright sunlight”; the other one says “light as bright as full daylight”. Does this difference in phrasing imply anything?

Can an undead creature like Strahd wield a Helm of Brilliance?

I had a thought that if my players don’t raid the treasury before their fight with Strahd, he would use the magic items inside during the final fight against the players. However…

The description for the Helm of Brilliance mentions that:

  • As long as it has at least one diamond, the helm emits dim light in a 30-foot radius when at least one undead is within that area. Any undead that starts its turn in that area takes 1d6 radiant damage.

Since wearing the helmet technically means being within the effects radius…

  • Does this mean Strahd would be damaged by the helm if he got anywhere near it, even if it was lying on the floor un-attuned?
  • Could he pluck out all the diamonds before the fight to eliminate this potential threat, or would it burn his hands or something similar?
  • Could any undead creature wear the helm of brilliance without it dealing damage to said wearer?