What is the role of product marketing in research? Do you write your questionnaires? Or just inform objectives etc.?

In my view, the goal is to 

1. help define the objectives of the research based on business & user needs 
2. support research as they execute the research plan to ensure the research methodology & questions deliver on the objectives 
3. Once research is complete, help narrow and refine how to apply the findings. I can't emphasize this one enough. We live in a world where data is everywhere. Most reports I see come back with 20+ pages of insights, and as a PMM, you're looking at results from multiple different sources. One of the most important things a PMM can do is help the business and PM narrow in on which insights are more critical to action.

It depends on the company and if you have a formal research function or not! I've done everything from recruit participants, write discussion guides, field surveys, moderate research, synthesize insights. Or, I've given a researcher objectives and key questions and then sat in on a few interviews here and there. I think every PMM should have the experience of running their own research at some point - it keeps you close to the customer, helps you empathize with them and develop relationships with strategic customers (like if you run a CAB). 

But, as an org grows, you won't always have the bandwdith to do so. So, you have to find more scalable ways of staying close to the customer via Gong calls or collaborating closely with Research teams to make it easy for you to listen in on research or calls if you want. It can be hard to prioritize in the day to day but will make you a stronger product marketer and able to add more value to your product, marketing and sales counterparts.