Are the Orders in opposition?
No. They are sworn brothers. The Order of the Phoenix were the theurgists and divine spell casters. The Order of the Dragon were the Knights and warriors.
Are they diametrically opposed (Good/Evil)(Law/Chaos)
The Order of the Phoenix were all Druids, some multiclass (Fochlucan Lyrist was popular) but the dominant class is Druid.
The Order of the Dragon were always melee classes, any are possible but Knight and Ranger were the most common.
Are there linking factors (race/alignment/religion) for the Orders?
Only duty. Both Orders were charged with keeping 'The Black King' sealed away.
Aside from that all those in the order are born with a mark (about 3"-4" in diameter).
The members of the Order of the Phoenix possess a birthmark that appears like a phoenix rising from its ashes.
The Order of the Dragon possess a mark that appears to be a side view of a dragon curled in the fetal position, the tip of the nose and the tail almost close the circle.
Could the strength of the Orders allow for dedicated Priestly spellcasters?
Certainly, all casters possessed Druid levels, other classes were acceptable providing the member remained true to the order. The order itself did not venerate any Deity, in fact all religions were welcome providing they kept the precepts of the Order in mind and they did not violate their Druid alignment restrictions (any neutral).
Do the Orders have Interplanar ties? Are they in opposition to Extraplanar forces?
No real ties. The gods pay them no heed what so ever, and certainly don't owe them any favors.
For sealing away 'The Black King' both orders were hunted to extinction by armies of Baatezu. Through out the millennia no one knows the orders ever existed save for a select few historians and secret societies.
Have the Orders strayed from their original ethos?
They performed their duties proudly, and then seemed to just vanish...
Yes and no. While the Orders accepted everyone who bore the mark (which could be of any alignment) no one was under any obligation to get along yet they all recognized they had a duty to perform: to destroy 'The Black King'
Can someone belong to both?
Technically The Order of the Phoenix and the Order of the Dragon were the same thing with two separate names to classify the difference between their professions (Order of the Dragon members never had Druid levels).
What are the obligations and goals of these two orders?
To destroy 'The Black King', however he was too powerful. The best they were able to do was seal his soul within a towering crystal and entomb it in the bottom of Lake Rytarr.
To ensure no one would shatter that seal they kept watch over the lake and its environs for several thousands of years. Nearly 1,000 years ago the last of the Orders (a Phoenix Druid, the Knights were slain first) fell to the Baatezu. Since his death, no others have been born with 'the mark'.
When you were born with the mark (one or the other) a representative of either order would find you and simply explain why you have it, and what your duty is now that you know its purpose. As simplistic as it sounds no one regardless of his or her alignment ever refused 'the call'.
Are these two orders well known or secret?
The knowledge of both orders is lost to all but the most learned of sages. The information that we work on here in this thread will act as a hand out to one of the players. Instead of letting them know everything I'm going to keep the info as lost as possible.