What are product marketing specific questions to ask on customer calls?

Three key questions that I would be trying to answer are:

1. What is the core kernel of value of this product/feature?
2. What problem is it and -- equally if not more important -- isn't it solving?
3. For which customers?

While these are important questions to answer, they are not questions you'd necessarily ask explicitly. For question 1, many customers won't be able to answer that straight up -- instead you'll glean it from how they answer a lot of the questions from PM and Design. Where there are gaps in your understanding, jump in with questions of your own. Questions 2 and 3 are answered with pattern recognition as you start to see trends in how customers speak about and interact with the product/feature. Start to develop hypotheses after the first few customer conversations and then test those hypotheses in later conversations.

Your main contribution won't be in driving a great conversation with customers, but instead a great conversation internally with your cross-functional partners. That's your chance to use the answers to 1-3 to influence your company's ability to achieve product/market fit and unreal real value for customers -- and consequently, real revenue for your business.

+1 to Gregg's answer. Not enough people do this. 

I think about this in terms of what are you listening for and how will you talk about it afterwards.

Product management: Do we need to change the way we do something?

Product design: Should it look or flow differently?

Product marketing: Why does the customer care? How will it save them time, money, toil, etc. What words do customers use to describe their pain? Is the customer pain an outlier or mainstream? How will they justify an investment to others?

For a current feature, I'd ask your customer about using it in a recent time period (eg last month or last week). How many times in that span? How did she use it, or even better show you how. What does she recall liking or disliking about it? For an upcoming feature, in recent time period, has there been a situation when she would have used it? What's an alternative way she completes that task now? Pros and cons?

It is critical to include Product Marketing in customer calls to understand customers' pain points, challenges, and the benefits they'd gain if they can remove their pain points. Product Marketing can then identify buyer/user personas, and business needs/challenges/use cases that span across customers across verticals, which can help finetune messaging, positioning, and sales enablement of the product while Product and Product Design teams can get feedback on the specific product features, capabilities, and UX/UI. Additionally, you'd build closer relationships with customers where you can ask for customer references or events in the future. 

First and foremost, it's fantastic that you've taken the initiative and made this a priority! Being in front of customers has been the single most important thing I've seen to drive a true understanding of our customers, and, perhaps even more important the challenges sales reps face day to day and how to build a better customer centric product with the right message to follow. In my past experiences, product marketing stated their purpose for being there in the introductions; I'm here to listen, learn and perhaps share and communicate our product and roadmap if applicable. Due to the early transparency and clarity for their attendence, the customer ended up looking at product marketing as a resource/consultant to their business. By removing pressure on product marketing to "inform" or "pitch" during the call they had conversations that left the customer and the PMM with critical takeaways. Those customers ended up having the most buy-in and built better long term relationships.