What is the best Legendary augment for a Monk in Diablo 3?
As for what items to augment first, it doesn't really matter, but in general you should augment items with better rolls. You are already using icy veins guides, so on this page Gear reccomentadion you can see desired rolls for all your items. If you have ancient items with good rolls - go for it, it doesn't matter if it's weapon or some other slot.
As for gem level, for season journey it's required to do augmentation of at least 50 level gem. Each gem level will give you 5 points of primary stat, so if you sacrifice level 100 gem you will receive bonus of 500 primary stat. The gem itself doesn't matter, you can use any legendary gem. Attribute is chosen by Flawless Royal gems which are required for recipe as well. For Monks, I think, dexterity is preferable stat, so use Flawless Royal Emeralds.
As for multiple augmentation, you can augment item more than once, but only the latest augmentation will be effective. High level augmentation makes sense only if you are going to push higher greater rifts.
Doesn't really matter, most people start with weapon and add their primary stat. For you that would be emeralds because of Dex.
The gem requirements are down below. It doesn't really matter what level it is, the higher obviously being better though. Don't kill yourself trying to get another 2 levels, it's not worth the time investment if you can't do it quickly.
No. Once you use the legendary Gem to upgrade an ancient item, you'll have to level a same legendary Gem from rank zero.
Legendary gem requirement:
Level 30+ for weapons
Level 40+ for rings and amulets
Level 50+ for armour including shields, quivers, and sources.