1) HQ materials only affect the initial quality. The more HQ materials, the higher the starting quality (up to 500, for all HQ materials).
Here's the exact calculation if you're feeling mathy
HQ/NQ mats
• Using all HQ mats will start you with 100 durability and 500 quality
• Using all NQ mats will start you with 100 durability and 0 quality
• Using a mashup of HQ/NQ will start you with 100 durability and 0~500 quality (see below for details)
Starting quality is weighted by optimal level of a material except when it is below 10. In that case, the optimal level is treated as 10 regardless of what it is.
InitialQuality = FLOOR(500 * (SumHQLevels / TotalLevels) )
SumHQLevels = Sum of HQ Material's Optimal Levels
TotalLevels = Sum of all Material's Optimal Levels
HQ rate
• HQ rate increases exponentially as your quality increases and is indicated at the bottom-right of the crafting window
• 300 quality = 1% HQ rate
• 500 quality = 7% HQ rate
• 750 quality = ~50% HQ rate
• 850 quality = ~75% HQ rate
• 1000 quality = 100% HQ rate
2) Skills only affect HQ chance by making your synth easier and allowing you to push your quality higher.
3) Sure, if you use them right, almost any would. Some are better than others.
4) The more crafts you level, the more abilities you have at your disposal. With the right set of abilities, HQing is much easier. Stacking the proper stat for your DoH + tool, will increase your quality gains.