First of all, desynth is totally different from regular synthing. I'll address the synthing first.
For more HQ chances, your best bet is either dark day or the day where the element is strong vs the crystal you are using (ex: Use fire crystal on waterday). New moon is also the best conditions for HQ. Of course, some do not believe in that, you are all entitled to your opinion. I personnally have seen a lot of improvement while following these rules.
One note, if you are 1-2 level above the recipe, it won't give you any more chance to get HQ... but why take a chance? I suggest you still follow it.
Here is a link where someone actually gathered some data to prove this:
http://ffxi.allakhazam.com/forum.html?forum=24&mid=1093897190388898652
Regarding the compass gibberish, I for one have always followed it since I discovered it and I have seen an improvement in my success rate and in HQ results. Again, this is purely an opinion thing and some crafters will tell you it's useless. Why then those the NPC shows you a map of the stars? Besides, why take any chances, it surely cannot hurt.
Now, desynth. Desynth is very much different then regular synthing. The main difference is that no matter your level, it always has a high risk of failure!! I'm almost to 100 alchemy and it still takes me 2-3 tries before desynthing a gob mask or blowing it up. On the other hand, if you succeed, it has a high chance of HQ result. For more info on desynth:
http://www.wickedways.lurazeda.com/viewtopic.php?t=81
I seriously doubt that the day or the moon phase affect the result of desynth, it ALWAYS have a high chance of failures. Maybe Full moon would be good, since you are trying to reverse the synthing process, but I have never tried it. But, as I said before, why take any chances? Try desynth on earth day (strong vs lightning), facing South and on a new moon, you'll see the result.
Finally, my suggestion to you would be to get all the furniture that gives Mog Enhancement: Desynth and try and get as many as you can. It won't improve the chances of success, but it reduces the risk of loosing the materials when desynthing.
Hope that helps and good luck.
EDIT: Just wanted to point out something, Light vs Dark. This has been a mistery for me still. If I am using a light crystal, should I wait for Dark day and face Dark for HQ results? Answer would be yes, as Dark is stronger over Light..... but then Light is also stronger over Dark, so in these conditions, one could also argue that I am doing this on the day weak against my crystal while facing the direction that my crystal s strong against.... think about it.
Now who uses dark crystal that could help us out.... Svina?
