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What are some good questions to ask during a second look? Is there anything off the table/inappropriate to ask?

Off the table? Not really (although still apply common sense, of course).

Suggestions that come to mind include what people would change about the firm if they could, how supportive the firm is of parenthood, opinions on the work assignment system, whether folks can count on getting their top practice group choice, what the culture is around working remotely, what typical hours are like, how much mentoring the partners provide.

I'd maybe ask about what the career trajectory looks like. What sorts of skills they hope to see out of a new attorney over the first couple years. Any tips and tricks for making life easier as a new attorney at the firm. Who do you need to know at the office in order to get things done.

Ask about exit options (where associates at the firm have exited to), ask how many people in your starting class are still at the firm (will give you an idea of attrition), billable requirements, whether there is bonus if you don't hit hours, if most people hit their hours or if most did not. Ask about an average day's schedule, how often they pulled late nighters, how often they work weekends. These are the things that really matter.

Keep in mind you will not get 100% honestly, these people will be trying to sell you.

Ask about the office support: ask about the support staff for the practice groups. Having good paras/adequate para support and others can be pretty damn important to avoid having a hellish experience.