Apparently sizing up the problem, Orky swam under the stretcher and allowed one of the men to stand on his head, something he’d never been trained to do.

Apparently sizing up the problem现在分词短语作状语，理解为时间状语比原因状语更为合理。After he apparently sized up the problem, Orky...

something he’d never been trained to do=which was something he'd never been trained to do. 如果是非限制性定语从句，先行词为主句的内容，是对主句作补充说明。which was省去后是一个名词（不定代词）+定语从句，是一个名词性短语，起的作用仍然是对主句的补充说明。我们不妨将这个名词短语成为句子的补充性状语。