Like Ashuckel said this is a good link to check.
But to answer your questions, Yes you can use those ores any time the upgrade level requirement is met (+10 HD Bradium/Carnium, +7 HD Oridecon/Elunium). The downgrade by 1 instead of breaking feature of HD ore is also always active with or without the events going on.
Also what you said is mostly right for normal Bradium/Carnium. But when they fail they can either downgrade by 3, even past the safe point of +10, or worst case they can critically fail and actually break/destroy the item.
The main drawback of attempting high level upgrades with HD Brad/Carn when Mad Bunny or Refine Rate event is not happening is that the base rate is half of what you get when those events are active. So if you're desperate it can still make sense to do +12 when an event is off but it is highly recommended not to try beyond +12 there as the success rate will be between 5-8%.
As for the lower level HD Ori/Elu they are not recommended to use at all these days though they can be used on normal (not shadow) equipment when it's over +7. If during your attempts using HD Elu/Ori the item fails and it goes to +6 you are no longer able to use the HD ore to continue.
For Enriched ore not during events it has no break or downgrade protection, so it is considered very risky. If the Enriched event is going on then items will get downgraded at that point. The only benefit of Enriched ore is the higher success rate so it pairs well with Blacksmiths Blessing if you need break protection and there is no event running.
Hope this helps and if you have more questions the fan wiki, linked above, is a great resource to check too.