Questions to ask a Headhunter who reached out via LinkedIn?

Hi all - a headhunter reached out via LinkedIn. I've always obtained roles through my network, so I'm green on what to ask them to maximize the quality of the interaction.

Here are the key points from their message to me, and some of my early questions. 

1. "You were confidentially referred to me as someone I should talk to about this role."

**I'd like to ask who referred me - is that typical to not cite who provided a referral? 

"My client is a fast growing family owned private investment firm looking for a CIO and COO across their investment portfolio of energy, infrastructure, and real estate."

**What are the right questions to ask about a family owned firm? I come from the KKR/Lazard world and have no experience in family offices, family owned companies - but I assume some fo the questions I'd ask are similar to what I'd ask if Blackstone was calling. 

"Firm are control focused but look to invest in situations of stress, instability and other special situations, and this includes flexibility up and down the cap stack across equity and credit."

**Hard to find stress out there in this market awash in liquidity...anyone disagree? 

"They are looking for a candidate that is looking for upward mobility and upside on the compensation."

**Is this code for accepting a low base salary / bonus and being locked into the fund?