What are the characteristics and themes present in Gundam characters based on Char Aznable?
It seems that there is always a masked antagonist in the Gundam series, but they vary in how much they resemble the original Char.

In Gundam Seed, it is Rau Le Creuset . Masked, and related by blood to a member of the protagonists camp. IIRC he is also a noble who stands to inherit, but unlike Char Aznable, 


   his family is not usurped.


In G-Gundam, it would be Schwarz Bruder. It is a bit tricky to bring him in the equation, though, because G-Gundam is much simpler than the other Gundam series.

Going by these examples and the ones mentioned in the question, the constant features seem to be


The character wears a mask to hide his true identity.
The character fights for the (apparent) enemy.
The character is male.
The character is related by blood or cloning to the protagonist or a close ally.


The features that vary among the Char Aznable knockoffs are


The character has a strong sense of honour and fair-play (IIRC Rau Le Creuset is the exception here).
The character is a noble.
The character bears a secret grudge (Schwarz Bruder would be the exception).
The character has a nickname (again Schwarz Bruder is the exception).


With respect to Quatre in Gundam Wing, I don't think he is a Char-based character. Quatre makes no secret of his identity and doesn't wear a mask. His name is not a clue IMO; almost all characters in the show were named after numbers in French. The role of Char is aptly fulfilled by Zechs Marquise.