Thanks to Plexus Du Menton, the dev of the game, we have the formula for equipment quality.
Important: This formula is for Boss equipment only

Simplify formula:
Quality = Max(10, (WorldTier * 10 + random(-5, 7.5) + zone * 1.5)) + Challenge SoulCoin Gain + CorruptionSlider + EquipmentQuality from soulshop + Expert's Monocle + Loop Scale

You can see your Equipment quality in your Stats page:


Important: Introduce in Hotfix 1.0.0.g we can have in some stages lower quality equipment which follow this formula minus a score we don't have

WorldTier*10:
This is your current run difficulty. if you launch a run in F difficulty it give you 10 points. 20 for E, 30 for D, etc...

random(-5, 7.5):
This is something calculated just before you drop an item, and this is a random value of 12.5 worth points. It can lower your quality or increase it.

Zone * 1.5:
This is the current zone you actually are in your run. If you are in zone 1 then you have 1.5 points, then you are zone 2 and you have 3 points, etc...
That mean the deeper you go in your run, the more point you have for equipment quality. That also mean going deeper in your run give you access to better quality equipment.
Important: In a loop, this parameter does not change and remains at the maximum value.

Challenge SoulCoin Gain:
Formula is : log10(Challenge SoulCoin Gain) * 5
This is a bonus for doing a challenge run. That mean if you are just doing a non challenge run you don't have a bonus for this part.
Endurance of Vild level 1 give you a bonus of 4.23 with its x7 soulcoin gain.

Endurance of Vild level 6 give you a bonus of 8.12 with its x42 soulcoin gain.

Because yes, some challenge can be level up and the more high they are the best will be your equipment.

CorruptionSlider:
Formula is : CorruptionSlider * 0.75 * (log400(Challenge SoulCoin Gain)
It's related to this slider, in the upper right of your starting run screen.

There is 11 graduation to this slider. This slider give you 4 bonus and when at max we have +200% soulcoins, +100% Talent Mastery, equipment quality of +8.25, and 500 corruption.

This is what we are talking about when the community are talking about 500 corruption run.
Important: We don't have any bonus quality points for any more corruption point we can get in a run. Even if you go to infinite corruption you won't get more equipment quality bonus. Only these 500 coming from this slider count.

EquipmentQuality from soulshop:
There is 3 different equipment quality from soulshop to buy.
Important: these bonus to equipment works even if a challenge limit our access to Soul-Shop.
C Tier, it give us up to 1.08 equipment quality bonus.

B Tier, it give us up to 4 equipment quality bonus.

A Tier, it give us up to 3.6 equipment quality bonus.


Loop Scale:
You need to break a mirror in an event to enable this feature.
Formula is : Loop Scale* log16(SoulCoin Multiplier)
After finishing a run you have the possibility to loop. The higher challenge level you do the higher your quality from Loop will be.
(You cannot loop again if an enemy has reached infinity health during a loop)

Expert's Monocle:
It's an unique equipment coming from Overgeared, a normal B Rank challenge.
It give us a nice flat bonus to quality depending of it's level and rank. Here at MAX-Tier level 10 because I am spending too much time on this game.

Now that we now how to calculate the quality here the table for the equipment tier we have corresponding of quality.

And that's not all, because the last digit of item quality +1 is equal to the level of our item.
For exemple if we drop an equipment of quality equal to 46 then we will have a C-Tier equipment of level 7.

Each modifier stats depend of equipment Tier and level.

It depend of the modifiers but often the formula is:

• For flat stat bonus:
  +5% of base stats per level, +50% of base stat per tier, when the modifier affect base/final stats.
  +1% of base multiplier per level, +10% of base multiplier per tier, when the modifier affect stats multiplier
  
• For stat per player level:
  +2.5% per level, +25% per tier, when the modifier affect base/final stats.
  +0.5% per level, +12.5% per tier, when the modifier affect stats multiplier on each level

For example, you have a piece of equipment that do + 2 power at F-rank level 1
It'll gain 0.1 power per level of equipment, and 1 power from level 10 F-rank to Level 1 E-rank

We only know about quality and stats progression, we have no possibility to know what kind of equipment will drop.