The odd opening position

XI looks vaguely like a variation of danish gambit on the king side shud be suicidal after f2+ for white

 BTW, the game is Schallopp-Spielmann,

http://www.chessgames.com/perl/chessgame?gid=1130652

I think IX could be English Opening: Anti-Benoni variation.

 No, but I will give you some hints:

1) Euwe is playing black

2) The pawn on c4 found itself there on white's third move.

Hint for XI. It is definitely King's Gambit accepted. You need to find a gambit variation.

i give up XI

IX is anti-sicilian

 No, I'll give another hint for IX: there is a player's surname in the variation, and it is neither Najdorf not Polugaevsky.

IX. Sicilian: O'Kelly Variation

XI. King's Gambit: Classical. Allgaier Variation

 Correct! Now only the main question remains!

I'd guess the odd one is no5?

Because of the fianchettoed bishop?

My bet would be on #2 because the black king leaves it's post and loses the right to castle, or #12 because it's the only one named after fictional characters.

Hint for the main question: the common shared characteristic is mathematical in nature.

Board #5 is the only game in which both players still have all 16 pieces.

Zank, good thinking, but not the characteristic I was looking for. You are on the right track, though, in terms of counting.

Well, you definitely win the award for the best puzzles in The Brainstrainers(and
maybe all of Chess.com).  My puzzles are pretty good but you have these great
graphics I can't replicate.

Board #1 is the only game with the queens off the board.

 True, but... not what I had in mind. The property is more mathematical, and if an openings expert looked at the positions he would probably say: "It looks as if the position xxx is different from the others, simply by comparing...." Enough said.

is Queen's Gambit Declined one only one in D00-D99?