Here is a question closely related to a characterization of standard atoms (not to be confused with standard alphabets:). It asks whether a bound on the number of automorphisms is equivalent to a bound on the numer of color-preserving automorphisms, in every Fraisse class of finite relational structures.
For a Fraisse class 
of finite structures over a finite relational vocabulary, are the following two conditions equivalent? Below, by a colored structure from we mean a structure from together with a mapping from its carrier to some finite set of colors.
A color class is the set of elements mapped to the same color.
1. The structures in have boundedly many automorphisms: for some number 
, every structure in has at most automorphisms.
2. The colored structures in have boundedly many automorphisms: for every number 
there is a number such that every colored structure in with color classes of size bounded by , has at most color-preserving automorphisms.
The following question seems vaguely related.
Are the following conditions equivalent for a Fraisse class
1. All its structures are rigid, i.e. have trivial automorphism groups
2. There is a definable linear order in the Fraisse limit.
The above equivalence holds for Ramsey classes. Any Ramsey class is a Fraisse class, but not the other way around. The equivalence for Ramsey classes is mentioned here. It seems highly nontrivial, and follows from the work of Kechris, Pestov, Tedorcevic.