HOW HQing WORKS
This doesn't really belong in a guide focused on grinding out levels, but since there are folks who don't seem to know the basics I figured it was about time to throw this explanation in here. This is information geared toward non crafters.

The kind of gear the crafter needs to hit those high numbers depends entirely on their playstyle, or what the person needing the synth wants them to do, whether they like high risk/high rewards or 100% success even if it doesn't HQ. There is no one right way to do it. There is also that pesky business of certain crafts, or certain synths within crafts (Ugh), needing Craftsmanship over Magic Craftsmanship and vice versa. Stacking the correct stat leads to more success. So a Goldsmith would stack as much Magic Craftsmanship as possible while an Armorsmith would stack Craftsmanship. Or in some cases when a person has multiple stat clashing crafts leveled they may balance all stats in order to keep their inventory from exploding with excess gear. All that said, insulting a crafter's gear choices when you are not paying them to do synths is a really good way to get them to not try for HQs for you. Keep that in mind.

The lower the level of the synth, the easier it is for an experienced crafter to hit 1000%. A crafter geared properly for their playstyle with good abilities equipped should be able to hit 870+ with no issues using HQ items for normal level 50 synths and below. Anything above level 50 is a little harder. Note that everything below 1000 (100%) is not 100%. Nothing else is guaranteed. You can hit 99% and still fail to HQ. The Double Down process cuts whatever % the crafter was at in half for one more shot to HQ and if it fails it breaks. Keep that in mind before you give the go ahead to Double Down and don't berate the crafter if you get unlucky. There isn't anything the crafter could have done about it.

The chance someone can synth an HQ item is based on the amount of quality the crafter can get before the synth finishes. Chances are pretty abysmal at first, if you use NQ items and start at 0, but start to skyrocket once you hit 750 quality and beyond. Here's a short list for those of you who like numbers:
~300 = 1% chance to HQ
~500 = 7% chance to HQ
~660 = 30% chance to HQ
~750 = 50% chance to HQ
~800 = 70% chance to HQ
~870 = 80% chance to HQ
~900 = 85% chance to HQ
~1000 = 100% chance to HQ
The amount of quality you start with is determined by how many materials are required to do the synth and how many of those materials are HQ. Having no HQ materials starts you at 0 quality (0%). Having all HQ materials, no matter how many materials are in the synth, equals 500 quality (7%) right off the bat, which is a huge boost in chance. There is currently no way to start above 500. But if you mix it up with NQ materials you get lower starting percentages. For example, if the synth takes 4 materials and only 2 are HQ then you will start with 250 quality (0%). If only 1 is HQ it would be 125 quality (0%). This is why bringing all or a majority of NQ mats, demanding an HQ and then crying about how HQ materials cost too much isn't going to net you much sympathy from the crafter.

tldr; If you want a better chance to get an HQ of the item you want, give the crafter as many HQ materials as is possible for the synth. Some synths you cannot have all HQ mats (the special class weapons being one). Even having all HQ items does not guarantee an HQ result, no matter how good the crafter is.