Hey Slapstick,
Not all of your questions were fully answered, so I'll add on with what I have learnt in the Far East:
Should I just burn through regions until one is found? How long should it take me to find one?
You should be able to find one in every region, just about. In perhaps 10 months, I can count on two hands the amount of times that I have not found a sage when looking. And I have looked for sages many times, and have made maybe 7 relics.
When I manage to find one, will I be able to collect the items required before they leave?
Most likely not. The item that they need will likely only be found in another region-type, and even if you are in the right region-type, odds are that you won't find the exact item that you need. You should either ask around for it to trade, or go to another region (the sages will ask for different things).
How expensive will these items be, and is there a way to predict what will be needed beforehand?
If you are gathering them, just depending on how long it takes. If you are trading, you will want to try provide a reasonable deal. As said before, trade your extras. I offer those items that I have 3 or more of, in general. Unfortunately due to "unknown factors" (to me, anyway), you cannot buy items from other adventurers. It is a dire lack. Depending on the damage, often the relic (unique item) will need more than one repair.
Some speculation: rarer objects seem to repair more than more common objects?