"Nice thing to say"—why not "thing nice to say"?

Because "thing" is a (pro)noun and "nice" is an adjective that modifies the noun: Postpositive adjectives are rare in English and their use is generally formulaic; "thing" and "nice" don't meet the criteria.

As for your second paragraph: this is a different situation. "Reading" in the first sentence is a verb - the boys are doing reading; in the second sentence "reading" is an adjective - "reading" is a feature of the boys. If instead we considered it a verb the sentence (as a sentence) would be incomplete - who is reading the boys under the trees?

I may be looking at this too simplistically, but it's the same reason we don't say, "she put on her dress red." Nice is an adjective modifying a noun and an adjective comes in front of the noun.

Aren't they different? "that is a nice thing to do!"-> the thing is nice. "that is a thing nice to do" -> doing the thing is nice.

A Google NGrams comparison thing_INF nice to,nice thing_INF to shows that, at the scale Google chooses, the usage of the forms "thing* nice to" is so rare that it doesn't show on the graph.

Compare instead question_INF hard to,hard question_INF to. Now Google NGrams shows that, though the forms "hard question* to" are used more often, the usage of the forms "question* hard to" at least shows up on the graph.

Perhaps one factor making "thing nice to..." so rare is that the noun "thing" carries very little semantic weight.