you got a break when the charges were reduced to disorderly conduct. A larceny conviction could have made you deportable and inadmissible and you might have been deported.
There is no difference between eligibility for a H-4 or a Green Card when it comes to crimes. Either you don't get a visa at all or you get one, no matter which one.
So since you were given the H-4, the criminal past should not be a problem to get a green card.
