Which abilities are inherited from previous dragons?

As of Patch X.31.8.0, hard mode and easy mode have been consolidated into one. The Ivory Wyvern is always fought, and the inheritance property of easy mode now applies in a different manner:

The first dragon killed will add environmental hazards to the following dragons’ arenas corresponding to their element
The second dragon killed will cause all the following ones to start using some of its shots
The third dragon killed will empower all of the following dragons’ minions with their element
The fourth dragon killed will assist the Ivory Wyvern in his fight as an invincible statue.

The wiki now has comprehensive information on each of the effects of the dragons, found in a handy little table.

I haven't found any sources for what the recommended order is, but in the 20-30 of these I've ran post-update, people have run Red first 80% of the time. I personally prefer running Blue first, as it's the safest option for all the other fights; feel free to look at the wiki page and make your own decision, though.