The two phrases that use "state government" are not accurate, because if you don't use the plural version of "government," you are referring to "government" as a concept, as in, "State government is an essential part of the way this country works." So you want to use the plural, because you are talking about compensation to any specific government of any specific state in need of this compensation; individual, possibly multiple, governments.
"For damaging" would imply that the entity giving the compensation was also the entity causing the damage, which may or may not be the case, in fact, but which is definitely not intended as part of the legislation. The legislation will give compensation for toxic chemical damage no matter where it originated or who caused it, and the legislation certainly does not intend to imply that the federal government caused it (whether they did or not).
"The giving of compensation" is unnecessarily elongated and pompous, and so is literally not incorrect, but would generally be discouraged because it is less clear, less direct, and less efficient.
"Damage of" is a nonstandard usage. "Damage to" is strongly preferred.
Therefore, none of your examples is really ideal, in my opinion. I would suggest that the first one comes closest, if you only pluralize "government" to "governments," resulting in "compensating state governments for damage to." (Other versions can work, but if we're sticking close to your examples, this is the best choice.)